# Modeling, Functions, and Graphs

### 1Functions and Their Graphs1.1Linear ModelsHomework 1.1

#### 1.

 $$h$$ $$0$$ $$3$$ $$6$$ $$9$$ $$10$$ $$T$$ $$65$$ $$80$$ $$95$$ $$110$$ $$115$$
1. $$\displaystyle T=65+5h$$
2. $$\displaystyle 95\degree$$
3. 3 p.m.

#### 3.

 $$w$$ $$0$$ $$4$$ $$8$$ $$12$$ $$16$$ $$A$$ $$250$$ $$190$$ $$130$$ $$70$$ $$10$$
1. $$\displaystyle A=250-15w$$
2. 75 gallons
3. Until the fifth week

#### 5.

1. $$\displaystyle P=-800+40t$$
2. $$(0,-800)\text{,}$$ $$(20,0)$$
3. The $$P$$-intercept, $$-800\text{,}$$ is the initial $$(t = 0)$$ value of the profit. Phil and Ernie start out $$\800$$ in debt. The $$t$$-intercept, $$20\text{,}$$ is the number of hours required for Phil and Ernie to break even.

#### 7.

1. $$\displaystyle C=5000+0.125d$$
2. Complete the table of values.
 Miles Driven $$4000$$ $$8000$$ $$12,000$$ $$16,000$$ $$20,000$$ Cost ($) $$5500$$ $$6000$$ $$6500$$ $$7000$$ $$7500$$ 3.$$$500$$
4. More than 16,000 miles

#### 13.

1. $$\displaystyle (8, 0), (0, 4)$$

#### 15.

1. $$\displaystyle (4, 0), (0, -3)$$

#### 17.

1. $$\displaystyle (9, 0), (0, -4)$$

#### 19.

1. $$\displaystyle (-2250, 0), (0, 1500)$$

#### 21.

1. $$\displaystyle (12, 0), (0, 4)$$

#### 23.

1. $$\displaystyle \left(\dfrac{3}{2} , 0\right), \left(0, \dfrac{11}{3} \right)$$

#### 25.

1. $$$2.40x,$$$$$3.20y$$
2. $$\displaystyle 2.40x + 3.20y = 19,200$$
3. The $$y$$-intercept, $$6000$$ gallons, is the amount of premium that the gas station owner can buy if he buys no regular. The $$x$$-intercept, $$8000$$ gallons, is the amount of regular he can buy if he buys no premium.

#### 27.

1. $$9x$$ mg, $$4y$$ mg
2. $$\displaystyle 9x + 4y = 1800$$
3. The $$x$$-intercept, $$200$$ grams, tells how much fig Delbert should eat if he has no bananas, and the $$y$$-intercept, $$450$$ grams, tells how much banana he should eat if he has no figs.

#### 29.

1. $$\displaystyle (3,0), (0,5)$$
2. $$\displaystyle \left(\dfrac{1}{2},0\right), \left(0,\dfrac{-1}{4}\right)$$
3. $$\displaystyle \left(\dfrac{5}{2},0\right), \left(0,\dfrac{-3}{2}\right)$$
4. $$\displaystyle (p,0), (0,q)$$
5. The value of $$a$$ is the $$x$$-intercept, and the value of $$b$$ is the $$y$$-intercept.

#### 31.

1. $$\displaystyle (0, b)$$
2. $$\left(\dfrac{-b}{m},0\right)\text{,}$$ if $$m\ne 0$$

#### 33.

$$-2x + 3y = 2400$$

#### 35.

$$3x + 400y = 240$$

#### 37.

1. $$\displaystyle y = 6 - 2x$$

#### 39.

1. $$\displaystyle y = \dfrac{3}{4}x-300$$

#### 41.

1. $$\displaystyle y = 0.02 - 0.04x$$

#### 43.

1. $$\displaystyle y = 210 - 35x$$

#### 45.

1. $$\displaystyle (100, 0), (0, 25)$$
2. $$\displaystyle y = 25 - \dfrac{1}{4}x$$

#### 47.

1. $$\displaystyle (0.04, 0), (0, -0.05)$$
2. $$\displaystyle y = 1.25x - 0.05$$

#### 49.

1. $$\displaystyle (-60, 0), (0, 12)$$
2. $$\displaystyle y = 12 + \dfrac{1}{5}x$$

#### 51.

1. $$\displaystyle (-42, 0), (0, -28)$$
2. $$\displaystyle y = \dfrac{-2}{3}x-28$$

### 1.2FunctionsHomework 1.2

#### 1.

Function; the tax is determined by the price of the item.

#### 3.

Not a function; incomes may differ for same number of years of education.

#### 5.

Function; weight is determined by volume.

#### 7.

Input: items purchased; output: price of item. Yes, a function because each item has only one price.

#### 9.

Input: topics; output: page or pages on which topic occurs. No, not a function because the same topic may appear in more than one page.

#### 11.

Input: studentsâ€™ names; output: studentsâ€™ scores on quizzes, tests, etc. No, not a function because the same student can have different grades on different tests.

#### 13.

Input: person stepping on scales; output: personâ€™s weight. Yes, a function because a person cannot have two different weights at the same time.

No

Yes

Yes

Yes

No

Yes

#### 27.

1. $$\displaystyle 60$$
2. $$\displaystyle 37.5$$
3. $$\displaystyle 30$$

#### 29.

1. $$\displaystyle 15\%$$
2. $$\displaystyle 14\%$$
3. $7010â€“$9169

#### 31.

1. $$67.7\text{:}$$ In 1985, $$67.7\%$$ of 20â€“24 year old women had not yet had children.
2. 1987: Approximately $$68\%$$ of 20â€“24 year old women had not yet had children in 1987.
3. $$\displaystyle f (1997) = 64.9$$

1. No
2. 60; no
3. Decreasing

1. 1991
2. 1 yr
3. 1 yr

2. 1989, about $$$5.10$$ 3. 1967, approximately 1970 #### 41. Answer. 1. $$\displaystyle 0$$ 2. $$\displaystyle 10$$ 3. $$\displaystyle -19.4$$ 4. $$\displaystyle \dfrac{14}{3}$$ #### 43. Answer. 1. $$\displaystyle 1$$ 2. $$\displaystyle 6$$ 3. $$\displaystyle \dfrac{3}{8}$$ 4. $$\displaystyle 96.48$$ #### 45. Answer. 1. $$\displaystyle \dfrac{5}{6}$$ 2. $$\displaystyle 9$$ 3. $$\displaystyle \dfrac{-1}{10}$$ 4. $$\displaystyle \dfrac{12}{13}\approx 0.923$$ #### 47. Answer. 1. $$\displaystyle \sqrt{12}$$ 2. $$\displaystyle 0$$ 3. $$\displaystyle \sqrt{3}$$ 4. $$\displaystyle \sqrt{0.2}\approx 0.447$$ #### 49. Answer. 1. $$V(12) = 1120\text{:}$$ After 12 years, the SUV is worth$$$1120\text{.}$$
2. $$t = 12.5\text{:}$$ The SUV has zero value after $$12\frac{1}{2}$$ years.
3. The value 2 years later

#### 33.

1. $$1250$$ barrels/day
2. The slope indicates that oil is pumped at a rate of $$1250$$ barrels per day.

#### 35.

1. $$-6$$ liters/day
2. The slope indicates that the water is diminishing at a rate of $$6$$ liters per day.

#### 37.

1. $$12$$ inches/foot
2. The slope gives the conversion rate of 12 inches per foot.

#### 39.

1. $$4$$ dollars/kilogram
2. The slope gives the unit price of $$\4$$ per kilogram

(a)

#### 43.

1. Yes, the slope between any two points is $$\frac{1}{2}\text{.}$$
2. $$0.5$$ grams of salt per degree Celsius

#### 45.

1. Yes
2. $$\displaystyle 2\pi$$

#### 47.

1. $$\displaystyle \dfrac{1500\text{ meters}}{1 \text{ second}}$$
2. $$3375$$ meters

#### 49.

1. The distances are known.
2. $$5.7$$ km per second

#### 51.

1. About $$18\degree$$C
2. 0.3 km to 0.4 km
3. About $$-28\degree$$C per kilometer

#### 53.

1. $$\displaystyle -3$$
2. $$\displaystyle 2$$

#### 55.

1. $$\displaystyle \dfrac{-1}{4}$$
2. $$\displaystyle -1$$

#### 57.

1. $$(1,F(1)),(4,F(4))\text{;}$$ $$~~~~F(4) - F(1)$$
2. $$(r,f(r)),(s,f(s))\text{;}$$ $$~~~~f(s) - f(r)$$

#### 59.

1. $$(2,H(2)),(3,H(3))\text{;}$$ $$~~~~H(3) - H(2)$$
2. $$(a,g(a)),(b,g(b))\text{;}$$ $$~~~~g(b) - g(a)$$

#### 61.

1. $$(c,s(c)),(d,s(d))\text{;}$$ $$~~~~s(c)(d - c)$$
2. $$(x_1,q(x_1)),(x_2,q(x_2))\text{;}$$ $$~~~~q(x_2)(x_2 - x_1)$$

#### 63.

1. $$(1, f (1)), (5, f (5))\text{;}$$ $$~~~~\dfrac{f (5) - f (1)}{4}$$
2. $$(-1, f (-1)), (2, f (2))\text{;}$$ $$~~~~\dfrac{f (2) - f (-1)}{3}$$

#### 65.

1. $$(a, f (a)), (b, f (b))\text{;}$$ $$~~~~\dfrac{f(b) - f(a)}{b-a}$$
2. $$(a, f (a)), (a+\Delta x, f(a+\Delta x))\text{;}$$ $$~~~~\dfrac{f(a+\Delta x) - f(a)}{\Delta x}$$

### 1.5Linear FunctionsHomework 1.5

#### 1.

1. $$\displaystyle y = \dfrac{1}{2}- \dfrac{3}{2}x$$
2. Slope $$\dfrac{-3}{2}\text{,}$$ $$y$$-intercept $$\dfrac{1}{2}$$

#### 3.

1. $$\displaystyle y = \dfrac{1}{9}- \dfrac{1}{6}x$$
2. Slope $$\dfrac{-1}{6}\text{,}$$ $$y$$-intercept $$\dfrac{1}{9}$$

#### 5.

1. $$\displaystyle y = -22 + 14x$$
2. Slope $$14\text{,}$$ $$y$$-intercept $$-22$$

#### 7.

1. $$\displaystyle y = -29$$
2. Slope $$0\text{,}$$ $$y$$-intercept $$-29$$

#### 9.

1. $$\displaystyle y =\dfrac{49}{3}-\dfrac{5}{3}x$$
2. Slope $$\dfrac{-5}{3}\text{,}$$ $$y$$-intercept $$\dfrac{49}{3}$$

#### 11.

1. $$\displaystyle y = -2 + 3x$$
2. $$\displaystyle \dfrac{2}{3}$$

#### 13.

1. $$\displaystyle y = -6 + \dfrac{5}{3}x$$
2. $$\displaystyle \dfrac{-18}{5}$$

#### 15.

$$5$$

#### 17.

$$\dfrac{-1}{4}$$

#### 19.

$$m =\dfrac{-A}{B}\text{,}$$ $$x$$-intercept $$\left(\dfrac{C}{A},0\right) \text{,}$$ $$y$$-intercept $$\left(0,\dfrac{C}{B}\right)$$

#### 21.

1. $$\displaystyle a = 100 + 150t$$
2. The slope tells us that the skierâ€™s altitude is increasing at a rate of $$150$$ feet per minute, the vertical intercept that the skier began at an altitude of $$200$$ feet.

#### 23.

1. $$\displaystyle G = 25 + 12.5t$$
2. The slope tells us that the garbage is increasing at a rate of $$12.5$$ tons per year, the vertical intercept that the dump already had $$25$$ tons (when the new regulations went into effect).

#### 25.

1. $$\displaystyle M = 7000 - 400w$$
2. The slope tells us that Tammyâ€™s bank account is diminishing at a rate of $$$400$$ per week, the vertical intercept that she had$$$7000$$ (when she lost all sources of income).

#### 27.

1. $$50\degree$$F
2. $$-20\degree$$C
3. The slope, $$\frac{9}{5} = 1.8\text{,}$$ tells us that Fahrenheit temperatures increase by $$1.8\degree$$ for each increase of $$1\degree$$ Celsius.
4. $$C$$-intercept $$\left(-17\frac{7}{9}, 0\right)\text{:}$$ $$-17\frac{7}{9}\degree$$ C is the same as $$0\degree$$F; $$F$$-intercept $$(0, 32)\text{:}$$ $$0\degree$$C is the same as $$32\degree$$F.

#### 29.

1. $$\displaystyle m = 25, ~b = 250$$
2. $$\displaystyle y = 250 + 25x$$

#### 31.

1. $$\displaystyle y = 0.12x + 25.4$$
2. $$18$$ kg

#### 33.

1. $$\displaystyle y + 5 = -3(x - 2)$$
2. $$\displaystyle y = 1 - 3x$$

#### 35.

1. $$\displaystyle y + 1 = \frac{5}{3}(x - 2)$$
2. $$\displaystyle y = \frac{-13}{3} + \frac{5}{3}x$$

#### 37.

1. $$\displaystyle y + 3.5 = -0.25(x + 6.4)$$
2. $$\displaystyle y = -5.1 - 0.25x$$

#### 39.

1. $$\displaystyle y + 250 = 2.4(x - 80)$$
2. $$\displaystyle y = -442 + 2.4x$$

#### 41.

1. $$\displaystyle m =\dfrac{2}{3}$$
2. $$\displaystyle y=\dfrac{-1}{3}+ \dfrac{2}{3}x$$

#### 43.

1. $$(-4, 4)\text{:}$$ neither; $$(0, 3)\text{:}$$ $$y = px + q\text{;}$$ $$(3, 2)\text{:}$$ both; $$(2, 1)\text{:}$$ neither; $$(1,-2)\text{:}$$ $$y = tx + v$$
2. $$p =\dfrac{-1}{3}\text{,}$$ $$q = 3\text{,}$$ $$t = 2\text{,}$$ $$v = -4$$

#### 45.

1. $$\displaystyle m = 4, ~b = 40$$
2. $$\displaystyle y = 40 + 4x$$

#### 47.

1. $$\displaystyle m = -80, ~b = -2000$$
2. $$\displaystyle P = -2000 - 80t$$

#### 49.

1. $$\displaystyle m = \dfrac{1}{4}, ~b = 0$$
2. $$\displaystyle V = \dfrac{1}{4}d$$

#### 51.

1. $$y = \dfrac{3}{4}x\text{,}$$ $$y = 1 + \dfrac{3}{4}x\text{,}$$ $$y = -2.7 + \dfrac{3}{4}x$$
2. The lines are parallel.

1. II
2. III
3. I
4. IV

1. III
2. IV
3. II
4. I

#### 57.

$$m = 2\text{;}$$ $$(6,-1)$$

#### 59.

$$m =\dfrac{-4}{3} \text{;}$$ $$(-5, 3)$$

#### 61.

1. The lines with slope $$3$$ and $$\frac{-1}{3}$$ are perpendicular to each other, and the lines with slope $$-3$$ and $$\frac{1}{3}$$ are perpendicular to each other.

#### 63.

$$m = -0.0018$$ degree/foot, so the boiling point drops with altitude at a rate of $$0.0018$$ degree per foot. $$b = 212\text{,}$$ so the boiling point is $$212\degree$$ at sea level (where the elevation $$h = 0$$).

### 1.6Linear RegressionHomework 1.6

#### 1.

1.  $$x$$ $$50$$ $$125$$ $$y$$ $$9000$$ $$15,000$$
2. $$\displaystyle C = 5000 + 80x$$
3. $$m = 80$$ dollars/bike, so it costs the company $$$80$$ per bike it manufactures. #### 3. Answer. 1.  $$g$$ $$12$$ $$5$$ $$d$$ $$312$$ $$130$$ 2. $$\displaystyle d = 26g$$ 3. $$m = 26$$ miles/gallon, so the Porcheâ€™s fuel efficiency is $$26$$ miles per gallon. #### 5. Answer. 1.  $$C$$ $$15$$ $$-5$$ $$F$$ $$59$$ $$23$$ 2. $$\displaystyle F=32+\dfrac{9}{5}C$$ 3. $$m =\dfrac{9}{5} \text{,}$$ so an increase of $$1\degree$$C is equivalent to an increase of $$\dfrac{9}{5}\degree$$F. #### 7. Answer. #### 9. Answer. #### 11. Answer. 1. $$12$$ seconds 2. $$\displaystyle 39$$ 3. $$11.6$$ seconds 4. $$\displaystyle y = 8.5 + 0.1x$$ 5. $$12.7$$ seconds; $$10.18$$ seconds; The prediction for the 40-year-old is reasonable, but not the prediction for the 12-year-old. #### 13. Answer. 1. $$\displaystyle y = 121 + 19.86t$$ 2. $$\displaystyle 419$$ #### 15. Answer. 1. $$\displaystyle y = 64.2 - 1.63t$$ 2. $$58$$ births per $$1000$$ women 3. $$32$$ births per $$1000$$ women #### 17. Answer. 1. $$\displaystyle y = 0.18t + 67.9$$ 2. $$74.9$$ years 3. $$79$$ years #### 19. Answer. 1. $$\displaystyle y = 90.49t - 543.7$$ 2. $$90.49$$ dollars/year: Each additional year of education corresponds to an additional$$$90.49$$ in weekly earnings.
3. No: The degree or diploma attained is more significant than the number of years. So, for example, interpolation for the years of education between a bachelorâ€™s and masterâ€™s degree may be inaccurate because earnings with just the bachelorâ€™s degree will not change until the masterâ€™s degree is attained. And the years after the professional degree will not add significantly to earnings, so extrapolation is inappropriate.

#### 21.

1. $$\displaystyle y = 1.6 + 0.11t$$
2. $$6.2$$ billion tons

#### 23.

1. $$0.34$$ meters per year
2. $$y = 0.34x$$ ($$b = 0$$ because the plant has zero size until it begins.)
3. Over $$1300$$ years

#### 25.

1. Yes
2. $$\displaystyle y = 1.29x - 1.62$$
3. The slope, $$1.29$$ kg/sq cm, tells us that strength increases by $$1.29$$ kg when the muscle cross-sectional area increases by $$1$$ sq cm.

#### 27.

1. E
2. $$y = 1.33x\text{;}$$ There should be no loss in mass when no gas evaporates.
3. $$1333$$ mg
4. Oxygen

#### 29.

1. $$75\degree$$F
2. The slope of $$-2$$ degrees/hour says that temperatures are dropping at a rate of $$2\degree$$ per hour.

#### 31.

1. $$20$$ mph
2. The slope of $$10$$ mph/second says the car accelerates at a rate of $$10$$ mph per second.

#### 33.

2 min: $$21\degree$$C; 2 hr: $$729\degree$$C; The estimate at 2 minutes is reasonable; the estimate at 2 hours is not reasonable.

#### 35.

$$128$$ lb.

#### 37.

1. $$\displaystyle y \approx -0.54x + 58.7$$
2. $$\displaystyle 31.7\%$$
3. $$90$$ meters
4. The regression line gives a negative probability, which is not reasonable.

#### 39.

1. $$\displaystyle y \approx 22.8x + 198.5$$
2. $$\approx 540$$ watts
3. $$198.5$$ watts
4. $$\approx -8.7$$ newtons
5. $$3.5$$ watts
6. about $$0.018$$ or $$1.8\%$$

### 1.7Chapter Summary and ReviewChapter 1 Review Problems

#### 1.

1.  $$n$$ $$100$$ $$500$$ $$800$$ $$1200$$ $$1500$$ $$C$$ $$4000$$ $$12,000$$ $$18,000$$ $$26,000$$ $$32,000$$
2. $$\displaystyle C = 20n + 2000$$
3. $$$22,000$$ 4. $$\displaystyle 400$$ #### 3. Answer. 1. $$\displaystyle R = 2100 - 28t$$ 2. $$(75, 0)\text{,}$$ $$(0, 2100)$$ 3. $$t$$-intercept: The oil reserves will be gone in 2080; $$R$$-intercept: There were $$2100$$ billion barrels of oil reserves in 2005. #### 5. Answer. 1. $$\displaystyle 2C + 5A = 1000$$ 2. $$(500, 0)\text{,}$$ $$(0, 200)$$ 3. $$C$$-intercept: If no adult tickets are sold, he must sell $$500$$ childrenâ€™s tickets; $$A$$-intercept: If no childrenâ€™s tickets are sold, he must sell $$200$$ adult tickets. #### 7. Answer. #### 9. Answer. #### 11. Answer. #### 13. Answer. #### 15. Answer. A function: Each $$x$$ has exactly one associated $$y$$-value. #### 17. Answer. Not a function: The IQ of $$98$$ has two possible SAT scores. #### 19. Answer. $$N(10) = 7000\text{:}$$ Ten days after the new well is opened, the company has pumped a total of $$7000$$ barrels of oil. #### 21. Answer. Function #### 23. Answer. Not a function #### 25. Answer. $$F(0) = 1, ~~F(-3) =\sqrt{37}$$ #### 27. Answer. $$h(8) = -6, ~~h(-8) = -14$$ #### 29. Answer. 1. $$\displaystyle f (-2) = 3, ~~f (2) = 5$$ 2. $$\displaystyle t = 1, ~~t = 3$$ 3. $$t$$-intercepts $$(-3, 0), (4, 0)\text{;}$$ $$f (t)$$-intercept: $$(0, 2)$$ 4. Maximum value of $$5$$ occurs at $$t = 2$$ #### 31. Answer. 1. $$\displaystyle x = \dfrac{1}{2}= 0.5$$ 2. $$\displaystyle x = \dfrac{27}{8}\approx 3.4$$ 3. $$\displaystyle x \gt 4.9$$ 4. $$\displaystyle x\le 2.0$$ #### 33. Answer. 1. $$\displaystyle x\approx\pm 5.8$$ 2. $$\displaystyle x = \pm 0.4$$ 3. $$-2.5\lt x \lt 0$$ or $$0\lt x\lt 2.5$$ 4. $$x\le -0.5$$ or $$x\ge 0.5$$ #### 35. Answer. $$H(2a) =4a^2 + 4a, ~~H(a+1) =a^2+4a+3$$ #### 37. Answer. $$f (a) + f (b) = 2a^2 + 2b^2 - 8, ~~f (a + b) = 2a^2 + 4ab + 2b^2 - 4$$ #### 39. Answer. The volleyball #### 41. Answer. Highway 33 #### 43. Answer. 1. $$\displaystyle B = 800 - 5t$$ 2. $$m = -5$$ thousand barrels/minute: The amount of oil in the tanker is decreasing by $$5000$$ barrels per minute. #### 45. Answer. 1. $$\displaystyle F = 500 + 0.10C$$ 2. $$m = 0.10\text{:}$$ The fee increases by $$\0.10$$ for each dollar increase in the remodeling job. #### 47. Answer. $$\dfrac{-3}{2}$$ #### 49. Answer. $$\dfrac{-34}{83}\approx-0.4$$ #### 51. Answer. $$80$$ ft #### 53. Answer. 1. $$\displaystyle h(x_2) - h(x_1)$$ 2. $$\displaystyle \dfrac{h(x_2) - h(x_1)}{x_2 - x_1}$$ #### 55. Answer. Neither #### 57. Answer.  $$d$$ $$V$$ $$-5$$ $$-4.8$$ $$-2$$ $$-3$$ $$1$$ $$-1.2$$ $$6$$ $$1.8$$ $$10$$ $$4.2$$ #### 59. Answer. $$m = \dfrac{1}{2}, ~b =\dfrac{-5}{4}$$ #### 61. Answer. $$m = -4, ~b = 3$$ #### 63. Answer. 1. $$\displaystyle y = \dfrac{10}{3}- \dfrac{2}{3}x$$ #### 65. Answer. 1. $$\displaystyle m = -2, ~b = 3$$ 2. $$\displaystyle y = 3 - 2x$$ #### 67. Answer. $$\dfrac{3}{5}$$ #### 69. Answer. 1. $$\displaystyle \dfrac{3}{2}$$ 2. $$(4,2)\text{,}$$ no 3. $$\displaystyle (6,5)$$ #### 71. Answer. $$(3,-14), ~(-7, 2)$$ #### 73. Answer. 1. $$\displaystyle T = 62 - 0.0036h$$ 2. $$-46\degree$$F; $$108\degree$$F 3. $$-71\degree$$F #### 75. Answer. $$y = \dfrac{2}{5}- \dfrac{9}{5}x$$ #### 77. Answer. 1.  $$t$$ $$0$$ $$15$$ $$P$$ $$4800$$ $$6780$$ 2. $$\displaystyle P = 4800 + 132t$$ 3. $$m = 132$$ people/year: the population grew at a rate of $$132$$ people per year. #### 79. Answer. $$6$$ #### 81. Answer. 1. $$129$$ lb, $$145$$ lb 2. $$\displaystyle y = 2.\overline{6} x - 44.\overline{3}$$ 3. $$137$$ lb 4. $$\displaystyle y=2.84x - 55.74$$ 5. $$137.33$$ lb #### 83. Answer. 1. $$45$$ cm 2. $$87$$ cm 3. $$\displaystyle y = 1.2x - 3$$ 4. $$69$$ cm 5. $$y = 1.197x - 3.660\text{;}$$ $$68.16$$ cm ### 2Modeling with Functions2.1Nonlinear ModelsHomework 2.1 #### 1. Answer. $$\pm\dfrac{5}{3}$$ #### 3. Answer. $$\pm\sqrt{6}$$ #### 5. Answer. $$\pm \sqrt{6}$$ #### 7. Answer. $$\pm 2.65$$ #### 9. Answer. $$\pm 5.72$$ #### 11. Answer. $$\pm 5.73$$ #### 13. Answer. $$\pm\sqrt{\dfrac{Fr}{m}}$$ #### 15. Answer. $$\pm\sqrt{\dfrac{2s}{g}}$$ #### 17. Answer. 1. $$\displaystyle V = 2.8 \pi r^2\approx 8.8r^2$$ 2.  $$r$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$V$$ $$8.8$$ $$35.2$$ $$79.2$$ $$140.7$$ $$219.9$$ $$316.7$$ $$431.0$$ $$563.0$$ The volume increases by a factor of $$4\text{.}$$ 3. $$5.86$$ cm #### 19. Answer. $$21$$ in. #### 21. Answer. $$\sqrt{1800}\approx 42.4$$ m #### 23. Answer. $$\sqrt{128}$$ in. by $$\sqrt{128}$$ in. $$\approx 11.3$$ in. $${}\times{} 11.3$$ in. #### 25. Answer. 1. $$\displaystyle x=\pm 12$$ #### 27. Answer. 1. $$x= 1$$ or $$x=9$$ #### 29. Answer. 1. $$x= 10$$ or $$x=-2$$ #### 31. Answer. $$5, -1$$ #### 33. Answer. $$\dfrac{5}{2}, \dfrac{-3}{2}$$ #### 35. Answer. $$-2 \pm \sqrt{3}$$ #### 37. Answer. $$\dfrac{1}{2} \pm \dfrac{\sqrt{3}}{2}$$ #### 39. Answer. $$\dfrac{-2}{9} , \dfrac{-4}{9}$$ #### 41. Answer. $$\dfrac{7}{8} \pm \dfrac{\sqrt{8}}{8}$$ #### 43. Answer. $$6$$ #### 45. Answer. $$64$$ #### 47. Answer. $$\dfrac{13}{6}$$ #### 49. Answer. $$8$$ #### 51. Answer. $$9$$ #### 53. Answer. $$\dfrac{33}{64}$$ #### 55. Answer. 1. $$\displaystyle B = 5000 (1 + r )^2$$ 2. $$\displaystyle 11.8\%$$ #### 57. Answer. $$8\%$$ #### 59. Answer. $$7.98$$ mm #### 61. Answer. 1. $$\sqrt{3}\approx 1.73$$ sq cm, $$4\sqrt{3}\approx 6.93$$ sq cm, $$25\sqrt{3}\approx 43.3$$ sq cm 2. An equilateral triangle with side $$5.1$$ cm has area $$11.263 \text{ cm}^2\text{.}$$ 3. $$\text{side}\approx 6.8$$ cm 4. $$\dfrac{\sqrt{3}}{4}s^2=20\text{;}$$ $$s \approx 6.8$$ 5. $$\approx 20$$ cm #### 63. Answer. $$\pm \sqrt{\dfrac{bc}{a}}$$ #### 65. Answer. $$a \pm 4$$ #### 67. Answer. $$\dfrac{-b\pm 3}{a}$$ #### 69. Answer. 1.  Height Base Area Height Base Area $$1$$ $$34$$ $$34$$ $$10$$ $$16$$ $$160$$ $$2$$ $$32$$ $$64$$ $$11$$ $$14$$ $$154$$ $$3$$ $$30$$ $$90$$ $$12$$ $$12$$ $$144$$ $$4$$ $$28$$ $$112$$ $$13$$ $$10$$ $$130$$ $$5$$ $$26$$ $$130$$ $$14$$ $$8$$ $$112$$ $$6$$ $$24$$ $$144$$ $$15$$ $$6$$ $$90$$ $$7$$ $$22$$ $$154$$ $$16$$ $$4$$ $$64$$ $$8$$ $$20$$ $$160$$ $$17$$ $$2$$ $$34$$ $$9$$ $$18$$ $$162$$ $$18$$ $$0$$ $$0$$ 2. $$162$$ sq ft, with base $$18$$ ft, height $$9$$ ft 3. Base: $$36 - 2x\text{;}$$ area: $$x (36 - 2x)$$ 4. See (a) 5. $$6.5$$ ft or $$11.5$$ ft #### 71. Answer. 1.  $$v$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$J$$ $$0$$ $$0.05$$ $$0.2$$ $$0.46$$ $$0.82$$ $$1.28$$ $$1.84$$ $$2.5$$ $$3.27$$ $$4.13$$ $$5.1$$ $$6.17$$ 2.  $$v$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$H$$ $$0.9$$ $$0.95$$ $$1.1$$ $$1.36$$ $$1.72$$ $$2.18$$ $$2.74$$ $$3.4$$ $$4.17$$ $$5.03$$ $$6.0$$ $$7.07$$ 3. $$5.5$$ meters 4. $$10.15$$ meters per second ### 2.2Some Basic FunctionsHomework 2.2 #### 1. Answer. 1. $$\displaystyle -9$$ 2. $$\displaystyle 9$$ #### 3. Answer. 1. $$\displaystyle -4$$ 2. $$\displaystyle 20$$ #### 5. Answer. $$-50$$ #### 7. Answer. $$144$$ #### 9. Answer. $$1$$ #### 11. Answer. 1. $$\displaystyle 2.7$$ 2. $$\displaystyle -2.7$$ 3. $$\displaystyle 1.8$$ 4. $$\displaystyle 2.9$$ #### 13. Answer. 1. $$\displaystyle 0.3$$ 2. $$\displaystyle -0.4$$ 3. $$\displaystyle 0.2$$ #### 15. Answer. 1. $$\displaystyle x = 0, ~x = 1$$ 2. $$(-\infty,0)$$ and $$(0,1)$$ #### 17. Answer. 1. $$\displaystyle x = 1$$ 2. $$\displaystyle (1, +\infty)$$ #### 19. Answer. Graph (b) is the basic graph shifted 2 units down; graph (c) is the basic graph shifted 1 unit up. #### 21. Answer. Graph (b) is the basic graph shifted 1.5 units left; graph (c) is the basic graph shifted 1 unit right. #### 23. Answer. Graph (b) is the basic graph reflected about the $$x$$-axis; graph (c) is the basic graph reflected about the $$y$$-axis. #### 25. Answer. 1. $$\displaystyle \sqrt{x}$$ 2. $$\displaystyle \sqrt[3]{x}$$ 3. $$\displaystyle \abs{x}$$ 4. $$\displaystyle \dfrac{1}{x}$$ 5. $$\displaystyle x^3$$ 6. $$\displaystyle \dfrac{1}{x^2}$$ #### 27. Answer. 1. $$\displaystyle x\approx 12$$ 2. $$\displaystyle x\approx 18$$ 3. $$\displaystyle x \lt 9$$ 4. $$\displaystyle x\gt 3$$ #### 29. Answer. 1. $$\displaystyle t\approx -3.1$$ 2. $$\displaystyle t\approx 1.5$$ 3. $$\displaystyle t \lt 0.8$$ 4. $$\displaystyle -2.4\lt t\lt 0.4$$ #### 31. Answer. 1. $$\displaystyle x = 41$$ 2. $$\displaystyle 29\lt x\lt 61$$ #### 33. Answer. 1. $$x = -5$$ or $$x=17$$ 2. $$\displaystyle -1\lt x\lt 13$$ #### 35. Answer. 1.  $$x$$ $$4$$ $$1$$ $$\frac{1}{4}$$ $$0$$ $$\frac{1}{4}$$ $$1$$ $$4$$ $$y$$ $$-2$$ $$-1$$ $$-\frac{1}{2}$$ $$0$$ $$\frac{1}{2}$$ $$1$$ $$2$$ 2. no #### 37. Answer. 1.  $$x$$ $$2$$ $$1$$ $$\frac{1}{2}$$ $$0$$ $$\frac{1}{2}$$ $$1$$ $$2$$ $$y$$ $$-2$$ $$-1$$ $$-\frac{1}{2}$$ $$0$$ $$\frac{1}{2}$$ $$1$$ $$2$$ 2. no #### 39. Answer. 1.  $$x$$ $$-\frac{1}{2}$$ $$-1$$ $$-2$$ undefined $$2$$ $$1$$ $$\frac{1}{2}$$ $$y$$ $$-2$$ $$-1$$ $$-\frac{1}{2}$$ $$0$$ $$\frac{1}{2}$$ $$1$$ $$2$$ 2. yes #### 41. Answer. #### 43. Answer. #### 45. Answer. #### 47. Answer. #### 49. Answer. #### 51. Answer. #### 53. Answer. $$f(x) = \begin{cases} 8-2x \amp x\lt 4\\ 2x-8 \amp x\ge 4 \end{cases}$$ #### 55. Answer. $$g(t) = \begin{cases} -1-\dfrac{t}{3} \amp t\lt -3\\ 1+\dfrac{t}{3} \amp t\ge -3 \end{cases}$$ #### 57. Answer. $$F(x) = \begin{cases} -x^3 \amp x\lt 0\\ x^3 \amp x\ge 0 \end{cases}$$ #### 59. Answer. 1. Not always true: $$f (1 + 2)\ne f (1) + f (2)$$ because $$9\ne 5\text{.}$$ 2. True: $$(ab)^2 = a^2b^2$$ #### 61. Answer. 1. Not always true: $$f (1 + 2)\ne f (1) + f (2)$$ because $$\frac{1}{3} \ne\frac{3}{2} \text{.}$$ 2. True: $$\dfrac{1}{ab} = \dfrac{1}{a}\cdot\dfrac{1}{b}$$ #### 63. Answer. 1. Not always true (unless $$b=0$$): $$f (1 + 2)\ne f (1) + f (2)$$ because $$3m+b \ne 3m+2b \text{.}$$ 2. Not always true: $$f (1\cdot 2)\ne f (1)\cdot f (2)$$ because $$2m+b \ne 2m^2 + 3mb + b^2 \text{.}$$ #### 65. Answer. #### 67. Answer. The distributive law shows a relationship between multiplication and addition that always holds. The equation $$f (a + b) = f (a) + f (b)$$ is not about multiplication and may or may not be true. ### 2.3Transformations of GraphsHomework 2.3 #### 1. Answer. $$y=\sqrt{x+2}$$ #### 3. Answer. $$y=x^3-1$$ #### 5. Answer. $$y=\dfrac{1}{x-4}$$ #### 7. Answer. 1. Translate $$y =\abs{x}$$ by $$2$$ units down. #### 9. Answer. 1. Translate $$y =\sqrt[3]{s}$$ by $$4$$ units right. #### 11. Answer. 1. Translate $$y =\dfrac{1}{t^2}$$ by $$1$$ unit up. #### 13. Answer. 1. Translate $$y =r^3$$ by $$2$$ units left. #### 15. Answer. 1. Translate $$y =\sqrt{d}$$ by $$3$$ units down. #### 17. Answer. 1. Translate $$y =\dfrac{1}{v}$$ by $$6$$ units left. #### 19. Answer. A vertical stretch by a factor of $$3\text{:}$$ $$y = \dfrac{3}{x}$$ #### 21. Answer. A vertical compression, the scale factor is $$\dfrac{1}{2}\text{:}$$ $$y = \dfrac{1}{2}x^3$$ #### 23. Answer. 1. Scale factor $$\frac{1}{3} \text{;}$$ $$y =\abs{x}$$ is compressed vertically by the scale factor. #### 25. Answer. 1. Scale factor $$-2 \text{;}$$ $$y =\frac{1}{z^2}$$ is reflected over the $$z$$-axis and stretched vertically by a factor of $$2\text{.}$$ #### 27. Answer. 1. Scale factor $$-3 \text{;}$$ $$y =\sqrt{v}$$ is reflected over the $$v$$-axis and stretched vertically by a factor of $$3\text{.}$$ #### 29. Answer. 1. Scale factor $$\frac{-1}{2} \text{;}$$ $$y =s^3$$ is reflected over the $$s$$-axis and compressed vertically by a factor of $$\frac{1}{2}\text{.}$$ #### 31. Answer. 1. Scale factor $$\frac{1}{3} \text{;}$$ $$y =\frac{1}{x}$$ is compressed vertically by the scale factor. #### 33. Answer. 1. vi 2. ii 3. iv 4. i 5. v 6. iii #### 35. Answer. 1. Vertical stretch by a factor of $$3\text{:}$$ $$y = 3 f (x)$$ 2. Reflection about the $$x$$-axis: $$y = -f (x)$$ 3. Translation $$1$$ unit right: $$y = f (x - 1)$$ 4. Translation $$4$$ units up: $$y = f (x) + 4$$ #### 37. Answer. 1. Reflection about the $$v$$-axis and vertical stretch by a factor of $$2\text{:}$$ $$T = -2h(v)$$ 2. Vertical stretch by a factor of $$3\text{:}$$ $$T = 3h(v)$$ 3. Translation $$3$$ units up: $$T = h(v) + 3$$ 4. Translation $$3$$ units left: $$T = h(v + 3)$$ #### 39. Answer. 1. Translation $$2$$ units up: $$y = f (x) + 2$$ 2. Translation $$4$$ units down: $$y = f (x) - 4$$ 3. Vertical compression by a factor of $$\frac{1}{2} \text{:}$$ $$y = \frac{1}{2}f (x)$$ 4. Translation $$1$$ unit right: $$y = f (x - 1)$$ #### 41. Answer. 1. Translation $$1$$ unit right: $$y = f (x - 1)$$ 2. Part (a) is translated $$30$$ units up: $$y = f (x - 1) + 30$$ 3. $$f$$ is reflected about the $$x$$-axis and stretched vertically by a factor of $$2\text{:}$$ $$y = -2 f (x)$$ 4. Part (c) is translated $$10$$ units down: $$y = -2 f (x) - 10$$ #### 43. Answer. $$y = \dfrac{1}{2}\cdot\dfrac{1}{x^2}$$ is a vertical compression with factor $$dfrac{1}{2}$$ of $$y = \dfrac{1}{x^2}\text{.}$$ #### 45. Answer. $$y = 2\sqrt[3]{x}$$ is a vertical stretch with factor $$2$$ of $$y = \sqrt[3]{x}\text{.}$$ #### 47. Answer. $$y = 3\abs{x}$$ is a vertical stretch with factor $$3$$ of $$y = \abs{x}\text{.}$$ #### 49. Answer. $$y = \dfrac{1}{8}x^3$$ is a vertical compression with factor $$\dfrac{1}{8}$$ of $$y = x^3\text{.}$$ #### 51. Answer. 1. Translation by $$2$$ units up and $$3$$ units right #### 53. Answer. 1. Translation by $$2$$ units left and $$3$$ units down. #### 55. Answer. 1. Reflection across the $$u$$-axis, vertical stretch by a factor of $$3\text{,}$$ translation by $$4$$ units left and $$4$$ units up #### 57. Answer. 1. Vertical stretch by a factor of $$2\text{,}$$ translation by $$5$$ units right and $$1$$ down #### 59. Answer. 1. Reflection across the $$w$$-axis, vertical stretch by a factor of $$2\text{,}$$ translation by $$6$$ units up and $$1$$ unit right #### 61. Answer. 1. Translation by $$8$$ units right and $$1$$ unit down #### 63. Answer. 1. Translation by $$4$$ units up and $$1$$ unit right: $$y = f (x - 1) + 4$$ 2. Vertical stretch by a factor of $$2$$ and a translation by $$4$$ units up: $$y = 2 f (x) + 4$$ #### 65. Answer. 1. $$y =\abs{x}$$ translated by $$1$$ unit left and $$2$$ units down 2. $$\displaystyle y =\abs{x+1} - 2$$ #### 67. Answer. 1. $$y =\sqrt{x}$$ reflected about the $$x$$-axis and shifted $$3$$ units up 2. $$\displaystyle y =-\sqrt{x} +3$$ #### 69. Answer. 1. $$y =x^3$$ translated by $$3$$ units right and $$1$$ unit up 2. $$\displaystyle y =(x - 3)^3 + 1$$ #### 71. Answer. 1. $$y = f(x - 20)\text{:}$$ Students scored $$20$$ points higher than Professor Hilbertâ€™s class. 2. $$y = 1.5 f(x)\text{:}$$ The class is about $$50\%$$ larger than Hilbertâ€™s, but the classes scored the same. #### 73. Answer. 1. $$y = f (x - 5000)\text{:}$$ Taxpayers earn$$$5000$$ more than Californians in each tax rate
2. $$y = f (x) - 0.2\text{:}$$ Taxpayers pay $$0.2\%$$ less tax than Californians on the same income.

#### 75.

1. $$y = g(t + 2)\text{:}$$ This population has its maximum and minimum two months before the marmots.
2. $$y = g(t) - 20\text{:}$$ This population remains $$20$$ fewer than that of the marmots.

### 2.4Functions as Mathematical ModelsHomework 2.4

(b)

(a)

(b)

1. II
2. IV
3. I
4. III

#### 19.

$$y = x^3$$ stretched or compressed vertically

#### 21.

$$y =\dfrac{1}{x}$$ stretched or compressed vertically

#### 23.

$$y =\sqrt{x}$$

1. Increasing
2. Concave up

#### 27.

1. Increasing
2. Concave down

#### 29.

1. Increasing, linear (neither concave up nor down)
2. C

#### 31.

1. Increasing, concave down
2. F

#### 33.

1. Decreasing, linear (neither concave up nor down)
2. D

#### 35.

$$y=4 \sqrt[3]{x}$$

#### 37.

$$y=3\cdot \dfrac{1}{x^2}$$

#### 39.

$$y=0.5 x^2$$

#### 41.

1. Table (4), Graph (C)
2. Table (3), Graph (B)
3. Table (1), Graph (D)
4. Table (2), Graph (A)

1. III
2. 3

#### 45.

1. $$\displaystyle S(x) = \begin{cases} 5.95 \amp x \le 25\\ 7.95 \amp 25\lt x\le 50\\ 9.95 \amp 50\lt x\le 75\\ 10.95 \amp 75\lt x\le 100 \end{cases}$$

#### 47.

1. During the first 400 seconds Bobâ€™s altitude is climbing with the aircraft; then the aircraft maintains a constant altitude of 10,000 feet for the next 100 seconds; after jumping from the plane, Bob falls for 20 seconds before opening the parachute; he falls at a constant rate after the chute opens.
2. $$240$$ seconds (4 minutes) and $$500 + \sqrt{250}\approx 515.8$$

#### 49.

1. $$m\approx 3.2$$ mm/cc: The height of precipitate increases by $$1$$ mm for each additional cc of lead nitrate
2. $$\displaystyle f(x)= \begin{cases} 1.34 + 3.2x \amp x \lt 2.6\\ 9.6 \amp x\ge 2.6 \end{cases}$$
3. The increasing portion of the graph corresponds to the period when the reaction was occurring, and the horizontal section corresponds to when the potassium iodide is used up.

1. II
2. IV
3. I
4. III

### 2.5The Absolute Value FunctionHomework 2.5

#### 1.

1. $$\displaystyle \abs{x}=6$$

#### 3.

1. $$\displaystyle \abs{p+3}=5$$

#### 5.

1. $$\displaystyle \abs{t-6}\lt 3$$

#### 7.

1. $$\displaystyle \abs{b+1}\ge 0.5$$

#### 9.

1. $$x = -5$$ or $$x = -1$$
2. $$\displaystyle -7\le x\le 1$$
3. $$x\lt -8$$ or $$x\gt 2$$

#### 11.

1. $$\displaystyle x = 4$$
2. No solution
3. No solution

#### 13.

$$x=\dfrac{-3}{2}$$ or $$x=\dfrac{5}{2}$$

#### 15.

$$q=\dfrac{-7}{3}$$

#### 17.

$$b=-14$$ or $$b=10$$

#### 19.

$$w=\dfrac{13}{2}$$ or $$w=\dfrac{15}{2}$$

No solution

No solution

#### 25.

$$\dfrac{-9}{2}\lt x \lt \dfrac{-3}{2}$$

#### 27.

$$d\le -2~$$ or $$~ d\ge 5$$

All real numbers

#### 31.

$$1.4 \lt t\lt 1.6$$

#### 33.

$$T\le 3.2~$$ or $$~T\ge 3.3$$

No solution

#### 37.

$$4.299\lt l\lt 4.301$$

#### 39.

$$250\le t\le 350$$

#### 41.

$$\abs{T - 5}\lt 0.3$$

#### 43.

$$\abs{D-100}\le 5$$

#### 45.

$$\abs{g-0.25}\le 0.001$$

#### 47.

1. $$\abs{t - 200}\lt 50\text{,}$$ $$~150\le t\lt 250$$
2. $$\abs{t - 200}\lt 0.5\text{,}$$ $$~199.5\le t\lt 200.5$$
3. $$\abs{t - 200}\lt 0.05\text{,}$$ $$~199.95\le t\lt 200.05$$

#### 49.

1. $$\displaystyle \abs{3x-6} = \begin{cases} -(3x-6) \amp \text{if } x\lt 2 \\ 3x-6 \amp \text{if } x\ge 2 \end{cases}$$
2. $$-(3x - 6)\le 9\text{,}$$ $$~3x - 6\lt 9$$
3. $$\displaystyle -1\lt x\lt 5$$
4. The solutions are the same.

#### 51.

1. $$\displaystyle \abs{2x+5} = \begin{cases} -(2x+5) \amp \text{if } x\lt \dfrac{-5}{2} \\ 2x+5 \amp \text{if } x\ge \dfrac{-5}{2} \end{cases}$$
2. $$-(2x+5)\gt 7\text{,}$$ $$~2x+5\gt 7$$
3. $$x\lt -6$$ or $$~x\gt 1$$
4. The solutions are the same.

#### 53.

1. $$\displaystyle f(x) = \begin{cases} -2x, \amp x\lt -4 \\ 8, \amp -4\le x\le 4 \\ 2x, \amp x\gt 4 \end{cases}$$
2. The graphs looks like like a trough. The middle horizontal section is $$y = p + q$$ for $$-p \le x\le q\text{,}$$ the left side, $$x\lt -p\text{,}$$ has slope $$-2$$ and the right side, $$x\gt q\text{,}$$ has slope $$2\text{.}$$
3. $$\displaystyle g(x) = \begin{cases} -2x+q-p, \amp x\lt -p \\ p+q, \amp -p\le x\le q \\ 2x+p-q, \amp x\gt q \end{cases}$$

#### 55.

1. $$\displaystyle f(x) = \begin{cases} -3x, \amp x\lt -4 \\ -x+8, \amp -4\le x\le 0 \\ x+8, \amp 0\lt x\lt 4 \\ 3x, \amp x\ge 4 \end{cases}$$
2. $$\displaystyle 8$$
3. $$\displaystyle p+q$$

#### 57.

1. $$\abs{x + 12}\text{,}$$ $$\abs{x + 4}\text{,}$$ $$\abs{x - 24}$$
2. $$\displaystyle f(x)=\abs{x + 12}+\abs{x + 4}+\abs{x - 24}$$
3. At $$x$$-coordinate $$-4$$

#### 59.

$$2$$ miles east of the river

### 2.6Domain and RangeHomework 2.6

#### 1.

Domain: $$[-5, 3]\text{;}$$ Range: $$[-3, 7]$$

#### 3.

Domain: $$[-4,5]\text{;}$$ Range: $$[-1, 1) \cup [3, 6]$$

#### 5.

Domain: $$[-2,2]\text{;}$$ Range: $$[-1,1]$$

#### 7.

Domain: $$(-5,5]\text{;}$$ Range: $$\{-1,0,2,3\}$$

#### 9.

1. Domain: all real numbers; Range: all real numbers
2. Domain: all real numbers; Range: $$[0, \infty)$$

#### 11.

1. Domain: all real numbers except zero; Range: $$(0, \infty)$$
2. Domain: all real numbers except zero; Range: all real numbers except zero

#### 13.

Domain: $$[0, 26.2]\text{;}$$ Range: $$[90, 300]$$

#### 15.

Domain: $$[0, 600]\text{;}$$ Range: $$[-90, 700]$$

#### 17.

1. $$\displaystyle V(t) = 6000 - 550t$$
2. Domain: $$[0, 10]\text{;}$$ Range: $$[500, 6000]$$

#### 19.

1. Domain: $$[0, 4]\text{;}$$ Range: $$[0, 64].~~$$The ball reaches a height of 64 feet and hits the ground 4 seconds after being hit.

#### 21.

1. Range: $$\{2.50, 2.90, 3.30, 3.70, 4.10\}$$

#### 27.

1.  $$A$$ $$10$$ $$100$$ $$1000$$ $$5000$$ $$10,000$$ $$S$$ $$25$$ $$42$$ $$69$$ $$98$$ $$115$$
2. $$81\text{,}$$ $$71$$
3. $$126,000$$ sq km

#### 29.

1. Home range size: II, lung volume: III, brain mass: I, respiration rate: IV
2. If $$p\gt 1\text{,}$$ the graph is increasing and concave up. If $$0\lt p\lt 1\text{,}$$ the graph is increasing and concave down. If $$p\lt 0\text{,}$$ the graph is decreasing and concave up.

#### 31.

1. Tricosanthes is the snake gourd and Lagenaria is the bottle gourd. Tricosanthes is thinner and Lagenaria is fatter.
2. $$\displaystyle a\approx 9.5$$
3. $$\displaystyle a\approx 2$$
4. Yes

#### 33.

1. $$79$$ species
2. $$18.4\degree$$C
3. $$f (9)\approx 85\text{,}$$ $$f (10)\approx 79\text{,}$$ $$f (19)\approx 49\text{,}$$ $$f (20)\approx 47\text{;}$$ from $$9\degree$$C to $$10\degree$$C has the greater decrease, corresponding to the steeper slope.

#### 35.

1. $$\displaystyle P=\dfrac{k}{\pi}d^{p-2}$$
2. The power function is a good fit on this interval.
3. $$\displaystyle 1.3$$

#### 37.

1. $$\displaystyle 4a^2$$
2. $$\displaystyle 9b^{5/3}$$

#### 39.

1. $$\displaystyle 4w^{3/2}$$
2. $$\displaystyle 3z^2$$

#### 41.

1. $$\displaystyle \dfrac{1}{2k^{1/4}}$$
2. $$\displaystyle \dfrac{4}{3h^{1/3}}$$

#### 43.

1. Wren: $$15$$ days, greylag goose: $$28$$ days
2. $$\displaystyle \dfrac{I(m)\cdot W(m)}{m}=0.18m^{-0.041}$$
3. Because $$m^{-0.041}$$ is close to $$m^0\text{,}$$ the fraction lost is close to $$0.18\text{.}$$

#### 45.

$$x = 64$$

#### 47.

$$x = \dfrac{1}{243}$$

#### 49.

$$x\approx 2.466$$

#### 51.

1. $$\displaystyle p= 1.115\times 10^{-12} a^{3/2}$$
2. $$1.88$$ years

#### 53.

$$\dfrac{13}{3}$$

#### 55.

$$0.665$$

#### 57.

$$2x^{3/2} - 2x$$

#### 59.

$$\dfrac{1}{2}y^{1/3}+\dfrac{3}{2}y^{-7/6}$$

#### 61.

$$2x^{1/2} - x^{1/4} - 1$$

#### 63.

$$a^{3/2}-4a^{3/4}+4$$

#### 65.

$$x(x^{1/2} + 1)$$

#### 67.

$$\dfrac{y-1}{y^{1/4}}$$

#### 69.

$$\dfrac{a^{2/3}+a^{1/3}-1}{a^{1/3}}$$

### 3.5Joint VariationHomework 3.5

#### 1.

1. $$\displaystyle R = f (x, y) = 129x + 240y$$
2. $$f (24, 12) = 5976$$ dollars is the maximum revenue.

#### 3.

1. $$f\left(4,\dfrac{3}{2}\right)=\dfrac{25}{12}\approx 2.1$$ inches
2. No: We do not have $$r = kL^2$$ for any constant $$k\text{.}$$
3. When $$L = 2r$$ and $$h = r\text{,}$$ $$f (L, h) = r\text{.}$$

#### 5.

1. $$16,220$$ sq cm
2. Height
3. $$\displaystyle 4.1\%$$

#### 7.

1. $$\displaystyle C = f (s,w)$$
2. $$f (4.5, 160) = 110\text{,}$$ so someone walking $$4.5$$ mph and weighing $$160$$ pounds burns $$110$$ calories per mile.
3. $$s\gt 4.5\text{.}$$ A person who weighs $$160$$ pounds must walk faster than $$4.5$$ mph in order to burn more than $$110$$ calories per mile.
4. $$7$$ mph
5. Find the row with your walking speed in the left column and move along that row until you are in the column with your weight at the top. The value in that row and column is the number of calories you burn per mile.

#### 9.

1. When is $$f (r, 20)\le 800\text{?}$$ $$~r \le7\%\text{;}$$ When is $$f (r, 30) \le 800\text{?}$$ $$~r\le 9\%$$
2. Reducing interest rate by $$5\%$$
3. No
4. No
5. $$30$$-year loan

#### 11.

1. Direct variation: In each row, $$E = km$$ for some constant $$k$$ that depends on the row.
2. Inverse variation: In each column, $$E = \dfrac{c}{g}$$ for some constant $$c$$ that depends on the column.
3. $$E = \dfrac{m}{g}$$ miles/gallon

#### 13.

1. In each row, $$R = kp$$ for some constant $$k$$ that depends on the row.
2. In each column, $$R = cd^2$$ for some constant $$c$$ that depends on the column.
3. $$\displaystyle R = 1.57d^2 p$$
4. $$588.75$$ pounds: When we keep $$p$$ constant and double $$d\text{,}$$ $$R$$ is multiplied by a factor of $$4\text{.}$$ So the value at $$d = 2\frac{1}{2} \text{,}$$ $$p = 60$$ should be $$4$$ times the value at $$d = 1\frac{1}{4} \text{,}$$ $$p = 60\text{.}$$

#### 15.

1. $$\displaystyle a=\dfrac{2d}{t^2}$$
2. Mercedes-Benz: $$19.05~ \text{ft}/\text{sec}^2\text{,}$$ Porsche: $$19.73~ \text{ft}/\text{sec}^2\text{,}$$ Dodge: $$20.47~ \text{ft}/\text{sec}^2\text{,}$$ Saleen: $$23.59~ \text{ft}/\text{sec}^2\text{,}$$ Ford: $$25.66~ \text{ft}/\text{sec}^2$$

#### 17.

1. $$\displaystyle L = \dfrac{3.2v^3}{R}$$
2. Increased by $$72.8\%$$
3. Decreased by $$16\frac{2}{3}\%$$

#### 19.

1.  Percent Ammonia Pressure (atmospheres) Temperature($$\degree$$C) $$50$$ $$100$$ $$150$$ $$200$$ $$250$$ $$300$$ $$350$$ $$400$$ $$350$$ $$25$$ $$38$$ $$46$$ $$53$$ $$58$$ $$62$$ $$66$$ $$68$$ $$400$$ $$16$$ $$26$$ $$33$$ $$38$$ $$45$$ $$48$$ $$53$$ $$56$$ $$450$$ $$9$$ $$17$$ $$23$$ $$28$$ $$32$$ $$37$$ $$40$$ $$43$$ $$500$$ $$6$$ $$11$$ $$16$$ $$20$$ $$23$$ $$27$$ $$29$$ $$32$$ $$550$$ $$4$$ $$8$$ $$11$$ $$14$$ $$17$$ $$19$$ $$22$$ $$24$$
2. The ammonia yield decreases.

### 3.6Chapter Summary and ReviewChapter 3 Review Problems

#### 1.

1. $$\displaystyle d = 1.75t^2$$
2. $$63$$ cm

#### 3.

$$480$$ bottles

#### 5.

1. $$\displaystyle w = \dfrac{k}{r^2}$$
2. $$3960\sqrt{3}\approx 6860$$ miles

#### 7.

$$y = 1.2x^2$$

#### 9.

$$y =\dfrac{20}{x}$$

#### 11.

1. $$\displaystyle \dfrac{1}{81}$$
2. $$\displaystyle \dfrac{1}{64}$$

#### 13.

1. $$\displaystyle \dfrac{1}{243m^5}$$
2. $$\displaystyle \dfrac{-7}{y^8}$$

#### 15.

1. $$\displaystyle \dfrac{2}{c^3}$$
2. $$\displaystyle \dfrac{99}{z^2}$$

#### 17.

1. $$\displaystyle 25\sqrt{m}$$
2. $$\displaystyle \dfrac{8}{\sqrt[3]{n}}$$

#### 19.

1. $$\displaystyle \dfrac{1}{\sqrt[4]{27q^3}}$$
2. $$\displaystyle 7\sqrt{u^3v^3}$$

#### 21.

1. $$\displaystyle 2x^{2/3}$$
2. $$\displaystyle \dfrac{1}{4}x^{1/4}$$

#### 23.

1. $$\displaystyle 6b^{-3/4}$$
2. $$\displaystyle \dfrac{-1}{3}b^{-1/3}$$

#### 29.

$$f(x)=\dfrac{2}{3}x^{-4}$$

#### 31.

1.  $$x$$ $$16$$ $$\dfrac{1}{4}$$ $$3$$ $$100$$ $$Q(x)$$ $$4096$$ $$\dfrac{1}{8}$$ $$4\sqrt{3^5}\approx 62.35$$ $$400,000$$

#### 33.

1.  $$x$$ $$0$$ $$1$$ $$5$$ $$10$$ $$20$$ $$50$$ $$70$$ $$100$$ $$f(x)$$ $$0$$ $$1$$ $$1.62$$ $$2.00$$ $$2.46$$ $$3.23$$ $$3.58$$ $$3.98$$

#### 35.

$$112$$ kg

1. 283
2. 2051

#### 39.

1. It is the cost of producing the first ship.
2. $$C = \dfrac{12}{ \sqrt[8]{x}}$$ million
3. About $$$11$$ million; about $$8.3\%$$ ; about $$8.3\%$$ 4. About $$8.3\%$$ #### 41. Answer. $$t=10$$ #### 43. Answer. $$x=7$$ #### 45. Answer. $$x=5$$ #### 47. Answer. $$x=75$$ #### 49. Answer. $$y=29,524$$ #### 51. Answer. $$g=\dfrac{2v}{t^2}$$ #### 53. Answer. $$p=\pm 2 \sqrt{R^2-R}$$ #### 55. Answer. $$49t^2$$ #### 57. Answer. $$\dfrac{k^7}{64}$$ #### 59. Answer. $$8a^2$$ #### 61. Answer. 1. $$132.6$$ km #### 63. Answer. 1. $$\displaystyle 480$$ 2. $$\displaystyle 498$$ #### 65. Answer. 1.$$$450$$
2. $$t = 8\text{:}$$ It costs $$$864$$ to insulate a ceiling with $$8$$ cm of insulation over an area of $$600$$ square meters. 3. $$\displaystyle C = 0.72A$$ 4. $$\displaystyle C = 18T$$ 5. $$\displaystyle C = 0.18AT$$ 6.$$$1440$$

#### 67.

1. $$N =\dfrac{k}{d^2E^3}\text{,}$$ where $$N$$ is number of people, $$d$$ is distance in miles from the road, $$E$$ is the elevation gain, and $$k$$ is the constant of variation.
2. $$\displaystyle k\approx 0.01$$
3. $$\displaystyle 3$$

### 4Exponential Functions4.1Exponential Growth and DecayHomework 4.1

#### 1.

1. $$$28$$ 2.$$$31.36$$

#### 3.

It is $$99\%$$ of what it was $$2$$ years ago.

#### 5.

1. $$P = 1200 + 150t\text{;}$$ $$1650$$
2. $$P = 1200\cdot 1.5^t\text{;}$$ $$4050$$

#### 7.

1. $$V = 18,000 - 2000t\text{;}$$ $$$8000$$ 2. $$V = 18,000\cdot 0.8^t\text{;}$$$$$5898.24$$

#### 9.

A: $$20\%\text{;}$$ B: $$2\%\text{;}$$ C: $$7.5\%\text{;}$$ D: $$100\%\text{;}$$ E: $$115\%$$

#### 11.

1. $$\displaystyle P = 20,000\cdot 2.5^{t/6}$$
2. $$36,840$$ bees; $$424,128$$ bees

#### 13.

1. $$\displaystyle A = 4000\cdot 1.08^t$$
2. $$$4665.60\text{;}$$$$$8635.70$$

#### 15.

1. $$\displaystyle P = 200,000\cdot 1.05^t$$
2. $$$359,171\text{;}$$$$$746,691$$

#### 17.

1. $$\displaystyle P = 250,000\cdot 0.75^{t/2}$$
2. $$162,380\text{;}$$ $$79,102$$

#### 19.

1. $$\displaystyle L = 0.85^{d/4}$$
2. $$44\%\text{;}$$ $$16\%$$

#### 21.

1. $$\displaystyle P = 50\cdot 0.992^t$$
2. $$46.1$$ lb; $$22.4$$ lb

#### 23.

1. $$\displaystyle 3^{x+4}$$
2. $$\displaystyle 3^{4x}$$
3. $$\displaystyle 12^x$$

#### 25.

1. $$\displaystyle b^{-2t}$$
2. $$\displaystyle b^{t/2}$$
3. $$\displaystyle 1$$

#### 27.

$$P (t + 1) = 12 (3)^{t+1}= 12 (3)^{t}\cdot 3= P(t) \cdot 3$$

#### 29.

$$P (x+k) = P_0 a^{x+k} = P_0 a^{x}\cdot a^k = P(x) \cdot a^k$$

#### 31.

1. In the expression $$2\cdot 3^t\text{,}$$ only the $$3$$ is raised to a power $$t\text{,}$$ and the result is doubled, but if both the $$2$$ and the $$3$$ were raised to the power $$t\text{,}$$ the result would be $$6^t\text{.}$$
2.  $$t$$ $$0$$ $$1$$ $$2$$ $$P(t)$$ $$2$$ $$6$$ $$18$$ $$Q(t)$$ $$1$$ $$6$$ $$36$$

#### 33.

$$4$$

#### 35.

$$1.2$$

#### 37.

$$r\approx 0.14$$

#### 39.

$$r\approx 0.04$$

#### 41.

1. $$\displaystyle P(t) = 1,545,387b^t$$
2. Growth factor $$1.049\text{;}$$ Percent rate of growth $$4.9\%$$
3. $$\displaystyle 2,493,401$$

#### 43.

1. $$\displaystyle 365$$
2. $$\displaystyle N(t) = 365(0.356)^t$$
3. $$0.03\text{.}$$ (Therefore, none)

#### 45.

The growth factor is $$1.5\text{.}$$
 $$t$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$P$$ $$~8~$$ $$12$$ $$18$$ $$27$$ $$40.5$$

#### 47.

The growth factor is $$1.2\text{.}$$
 $$x$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$Q$$ $$20$$ $$24$$ $$28.8$$ $$34.56$$ $$41.47$$

#### 49.

The decay factor is $$0.8\text{.}$$
 $$w$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$N$$ $$120$$ $$96$$ $$76.8$$ $$61.44$$ $$49.15$$

#### 51.

The decay factor is $$0.8\text{.}$$
 $$t$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$C$$ $$10$$ $$8$$ $$6.4$$ $$5.12$$ $$4.10$$

#### 53.

The growth factor is $$1.1\text{.}$$
 $$n$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$B$$ $$200$$ $$220$$ $$242$$ $$266.2$$ $$292.82$$

#### 55.

1. Initial value $$4\text{,}$$ growth factor $$2^{1/3}$$
2. $$\displaystyle f (x) = 4\cdot 2^{x/3}$$

#### 57.

1. Initial value $$80\text{,}$$ decay factor $$\frac{1}{2}$$
2. $$\displaystyle f (x) = 80\cdot \left(\dfrac{1}{2} \right)^x$$

#### 59.

$$84.6\%\text{,}$$ $$55.8\%$$

#### 61.

No, an increase of $$48\%$$ in $$6$$ years corresponds to a growth factor of $$1.48^{1/6}\approx 1.0675\text{,}$$ or an annual growth rate of about $$6.75\%\text{.}$$

#### 63.

1. $$\displaystyle P(t) = 16,986,335(1 + r)^t$$
2. $$\displaystyle 2.07\%$$

#### 65.

1. $$\displaystyle 3.53\%$$
2. $$\displaystyle 3.53\%$$
3. No
4. $$\displaystyle 3.53\%$$

#### 67.

1. $$39\text{;}$$ $$1.045$$
2. $$35\text{;}$$ $$1.047$$
3. Species B

#### 69.

1.  $$t$$ $$0$$ $$2$$ $$4$$ $$6$$ $$8$$ $$L(t)$$ $$3$$ $$6$$ $$9$$ $$12$$ $$15$$
$$\displaystyle L(t) = 3 + 1.5t$$
2.  $$t$$ $$0$$ $$2$$ $$4$$ $$6$$ $$8$$ $$E(t)$$ $$3$$ $$6$$ $$12$$ $$24$$ $$48$$
$$\displaystyle E(t) = 3\cdot 2^{t/2}$$

#### 71.

1. $$244$$ tigers per year
2. $$0.97\text{;}$$ $$3\%$$
3. Linear: $$3067\text{;}$$ Exponential: $$4170$$

### 4.2Exponential FunctionsHomework 4.2

#### 1.

1. $$26\text{;}$$ increasing
2. $$1.2\text{;}$$ decreasing
3. $$75\text{;}$$ decreasing
4. $$\frac{2}{3} \text{;}$$ increasing

#### 3.

 $$x$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$f(x)=3^x$$ $$\frac{1}{27}$$ $$\frac{1}{9}$$ $$\frac{1}{3}$$ $$1$$ $$3$$ $$9$$ $$27$$ $$g(x)=\left(\frac{1}{3} \right)^x$$ $$27$$ $$9$$ $$3$$ $$1$$ $$\frac{1}{3}$$ $$\frac{1}{9}$$ $$\frac{1}{27}$$
The two graphs are reflections of each other across the $$y$$-axis. $$f$$ is increasing, $$g$$ is decreasing. $$f$$ has the negative $$x$$-axis as an asymptote, and $$g$$ has the positive $$x$$-axis as its asymptote.

#### 5.

 $$t$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$h(t)=4^{-t}$$ $$64$$ $$16$$ $$4$$ $$1$$ $$\frac{1}{4}$$ $$\frac{1}{16}$$ $$\frac{1}{64}$$ $$q(t)=-4^t$$ $$\frac{-1}{64}$$ $$\frac{-1}{16}$$ $$\frac{-1}{4}$$ $$-1$$ $$-4$$ $$-16$$ $$-64$$
The graphs are reflections of each other across the origin. Both are decreasing, but $$h$$ has the negative $$t$$-axis as an asymptote, and $$q$$ has the positive t-axis as its asymptote.

1. I
2. IV
3. III
4. II

#### 9.

1. $$\displaystyle [1.08, 14.85]$$

#### 11.

1. $$\displaystyle [16.38, 152.59]$$

#### 13.

Because they are defined by equivalent expressions, (b), (c), and (d) have identical graphs

#### 15.

1. To evaluate $$f$$ we subtract $$1$$ from the input before evaluating the exponential function; to evaluate $$g$$ we subtract $$1$$ from the output of the exponential function.
2.  $$x$$ $$y=2^x$$ $$f(x)$$ $$g(x)$$ $$-2$$ $$\dfrac{1}{4}$$ $$\dfrac{1}{8}$$ $$\dfrac{-3}{4}$$ $$-1$$ $$\dfrac{1}{2}$$ $$\dfrac{1}{4}$$ $$\dfrac{-1}{2}$$ $$0$$ $$1$$ $$\dfrac{1}{2}$$ $$0$$ $$1$$ $$2$$ $$1$$ $$1$$ $$2$$ $$4$$ $$2$$ $$3$$
3. The graph of $$f$$ is translated $$1$$ unit to the right; the graph of $$g$$ is shifted $$1$$ unit down.

#### 17.

1. To evaluate $$f$$ we take the negative of the output of the exponential function; to evaluate $$g$$ we take the negative of the input.
2.  $$x$$ $$y=3^x$$ $$f(x)$$ $$g(x)$$ $$-2$$ $$\dfrac{1}{9}$$ $$\dfrac{-1}{9}$$ $$9$$ $$-1$$ $$\dfrac{1}{3}$$ $$\dfrac{-1}{3}$$ $$3$$ $$0$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$3$$ $$-3$$ $$\dfrac{1}{3}$$ $$2$$ $$9$$ $$-9$$ $$\dfrac{1}{9}$$
3. The graph of $$f$$ is reflected about the $$x$$-axis; the graph of $$g$$ is reflected about the $$y$$-axis.

#### 19.

1. $$3(5^{a+2})$$ is not equivalent to $$9\cdot 3(5^a)\text{.}$$
2. $$3(5^{2a})$$ is not equivalent to $$2\cdot 3 (5^a)\text{.}$$

#### 21.

1. $$8^w - 8^z$$ is not equivalent to $$8^{w-z}\text{.}$$
2. $$8^{-x}$$ is equivalent to $$\dfrac{1}{8^x}\text{.}$$

#### 23.

1. $$\displaystyle P_0=300$$
2.  $$x$$ $$0$$ $$1$$ $$2$$ $$f(x)$$ $$300$$ $$600$$ $$1200$$
3. $$\displaystyle b=2$$
4. $$\displaystyle f(x)=300(2)^x$$

#### 25.

1. $$\displaystyle S_0=150$$
2. $$\displaystyle b\approx 0.55$$
3. $$\displaystyle S(d) = 150(0.55)^d$$

#### 27.

$$\dfrac{2}{3}$$

#### 29.

$$\dfrac{-1}{4}$$

#### 31.

$$\dfrac{1}{7}$$

#### 33.

$$\dfrac{-5}{4}$$

#### 35.

$$\pm 2$$

#### 37.

1. $$\displaystyle N(t) = 26(2)^{t/6}$$
2. $$72$$ days later

#### 39.

1. $$\displaystyle V(t) = 700(0.7)^{t/2}$$
2. $$4$$ yr

#### 41.

$$x = 2.26$$

#### 43.

$$x = -1.40$$

1. Power
2. Exponential
3. Power
4. Neither

#### 47.

1. Exponential $$y=3\cdot 2^x$$
2. Power $$P=0.5 t^2$$

#### 49.

1. Power $$y=100 x^{-1}$$
2. Exponential $$P=\frac{1}{4} \cdot 2^x$$

#### 51.

 $$x$$ $$f(x)=x^2$$ $$g(x)=2^x$$ $$-2$$ $$4$$ $$\frac{1}{4}$$ $$-1$$ $$1$$ $$\frac{1}{2}$$ $$0$$ $$0$$ 1 $$1$$ $$1$$ $$2$$ $$2$$ $$4$$ $$4$$ $$3$$ $$9$$ $$8$$ $$4$$ $$16$$ $$16$$ $$5$$ $$25$$ $$32$$
1. Range of $$f\text{:}$$ $$[0, \infty)\text{;}$$ Range of $$g\text{:}$$ $$(0, \infty)$$
2. $$\displaystyle 3$$
3. $$-0.7667\text{,}$$ $$2\text{,}$$ $$4$$
4. $$(-0.7667, 2)$$ and $$(4,\infty)$$
5. g

#### 53.

1. $$y = 3^x - 4$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(-4, \infty)\text{,}$$ $$x$$-intercept $$(1.26, 0)\text{;}$$ $$y$$-intercept $$(0, -3)\text{;}$$ horizontal asymptote $$y=-4$$
2. $$y=3^{x-4}\text{,}$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(0, \infty)\text{,}$$ no $$x$$-intercept; $$y$$-intercept $$\left(0, \dfrac{1}{81}\right)\text{;}$$ the $$x$$-axis is the horizontal asymptote.
3. $$y=-4\cdot 3^{x}\text{,}$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(-\infty, 0)\text{,}$$ no $$x$$-intercept; $$y$$-intercept $$(0, -4)\text{;}$$ the $$x$$-axis is the horizontal asymptote.

#### 55.

1. $$y =-6^t$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(-\infty, 0)\text{,}$$ no $$t$$-intercept; $$y$$-intercept $$(0, -1)\text{;}$$ the $$t$$-axis is the horizontal asymptote.
2. $$y=6^{-t}\text{,}$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(0, \infty)\text{,}$$ no $$t$$-intercept; $$y$$-intercept $$(0, 1)\text{;}$$ the $$t$$-axis is the horizontal asymptote.
3. $$y=-6^{-t}\text{,}$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(-\infty, 0)\text{,}$$ no $$t$$-intercept; $$y$$-intercept $$(0, -1)\text{;}$$ the $$t$$-axis is the horizontal asymptote.

#### 57.

1. $$y =2^{x-3}$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(0, \infty)\text{,}$$ no $$x$$-intercept; $$y$$-intercept $$\left(0, \frac{1}{8}\right)\text{;}$$ the $$x$$-axis is the horizontal asymptote.
2. $$y=2^{x-3}+4 \text{,}$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(4, \infty)\text{,}$$ no $$x$$-intercept; $$y$$-intercept $$>\left(0, \frac{33}{8}\right)\text{;}$$ horizontal asymptote $$y=4$$

#### 59.

1. $$y =-\left(\dfrac{1}{2}\right)^t$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(-\infty, 0)\text{,}$$ no $$t$$-intercept; $$y$$-intercept $$(0, -1)\text{;}$$ the $$t$$-axis is the horizontal asymptote.
2. $$y=6-\left(\dfrac{1}{2} \right)^t \text{,}$$
Domain: $$(-\infty, \infty)\text{;}$$ range: $$(-\infty,6)\text{,}$$ $$t$$-intercept approximately $$(-2.58,0)\text{;}$$ $$y$$-intercept $$(0, 5)\text{;}$$ horizontal asymptote is $$y=6$$

#### 61.

1. The graph of $$y = 2^x$$ has been reflected about the $$y$$-axis and shifted up $$2$$ units.
2. $$\displaystyle y=2^{-x}+2$$

#### 63.

1. The graph of $$y = 2^x$$ has been reflected about the $$x$$-axis and shifted up $$10$$ units.
2. $$\displaystyle y=-2^{x}+10$$

1. I
2. III
3. II

#### 67.

1.  $$t$$ $$3.5$$ $$4$$ $$8$$ $$10$$ $$15$$ $$f(t)$$ $$128$$ $$154.75$$ $$184.05$$ $$150.93$$ $$103.96$$
2. From $$0$$ to $$3$$ minutes, the volunteer is walking with heart rate $$100$$ beats per minute. The volunteer jogged at a steady pace from $$3$$ to $$4$$ minutes, and the heart rate increased to about $$155$$ beats per minutes. From $$4$$ to $$9$$ minutes, the jogging pace increased, and the heart rate rose to about $$185$$ beats per minute. The cooldown started at $$9$$ minutes, and the heart rate decreased rapidly and leveled off to about $$100$$ beats per minute.

### 4.3LogarithmsHomework 4.3

#### 1.

1. $$\displaystyle 2$$
2. $$\displaystyle 5$$

#### 3.

1. $$\displaystyle \dfrac{1}{2}$$
2. $$\displaystyle -1$$

#### 5.

1. $$\displaystyle 1$$
2. $$\displaystyle 0$$

#### 7.

1. $$\displaystyle 5$$
2. $$\displaystyle 6$$

#### 9.

1. $$\displaystyle -1$$
2. $$\displaystyle -3$$

#### 11.

$$\log_2 1024=10$$

#### 13.

$$\log_{10} 5\approx 0.699$$

#### 15.

$$\log_{t} 16=\dfrac{3}{2}$$

#### 17.

$$\log_{0.8} M=1.2$$

#### 19.

$$\log_{x} (W-3)=5t$$

#### 21.

$$\log_{3} (2N_0)=-0.2t$$

#### 23.

1. $$\displaystyle \log_4 2.5$$
2. $$\displaystyle 0.7$$

#### 25.

1. $$\displaystyle \log_{10} 0.003$$
2. $$\displaystyle -2.5$$

#### 27.

1. $$\displaystyle 0\lt \log_{10} 7 \lt 1$$
2. $$\displaystyle 0.85$$

#### 29.

1. $$\displaystyle 3\lt \log_{3} 67.9 \lt 4$$
2. $$\displaystyle 3.84$$

#### 31.

1. $$\displaystyle 0.7348$$
2. $$\displaystyle 1.7348$$
3. $$\displaystyle 2.7348$$
4. $$\displaystyle 3.7348$$
When the input to the common logarithm is multiplied by $$10\text{,}$$ the output is increased by $$1\text{.}$$

#### 33.

1. $$\displaystyle 0.3010$$
2. $$\displaystyle 0.6021$$
3. $$\displaystyle 0.9031$$
4. $$\displaystyle 1.2041$$
When the input to the common logarithm is doubled, the output is increased by about $$0.3010\text{.}$$

#### 35.

$$-0.23$$

#### 37.

$$2.53$$

#### 39.

$$0.77$$

#### 41.

$$-0.68$$

#### 43.

$$3.63$$

#### 45.

$$2\cdot 5^x\ne 10^x\text{;}$$ the first step should be to divide both sides of the equation by $$2\text{;}$$ $$x = \log_5 424\text{.}$$

#### 47.

$$\frac{10^{4x}}{4} \ne 10^x\text{;}$$ the first step should be to write $$4x = \log 20\text{;}$$ $$x = \frac{\log 20}{4}\text{.}$$

#### 49.

1. $$\displaystyle 33,855,812$$
2. $$38,515,295\text{;}$$ $$~41,080,265\text{;}$$ $$~43,816,051$$
3. $$\displaystyle 2002$$
4. $$\displaystyle 2012$$

#### 51.

1. $$\displaystyle 85.5$$
2. Decreasing; range: $$[5.4, 1355.2]$$
3. $$\displaystyle 1.45$$
4. $$\displaystyle \dfrac{1}{100}$$
5. $$\displaystyle 10^{0.4}\approx 2.5119$$
6. $$2.15\times 10^{-6}$$ to $$855,067$$

#### 53.

$$9.60$$ in

#### 55.

$$1.91$$ mi

#### 57.

$$3.34$$ mi

#### 59.

$$1$$

#### 61.

$$0$$

#### 63.

$$1$$

#### 65.

$$0$$

### 4.4Properties of LogarithmsHomework 4.4

#### 1.

1. $$\displaystyle 10^8$$
2. $$2\text{;}$$ $$~6\text{;}$$ $$~8\text{;}$$ $$~2+6=8$$

#### 3.

1. $$\displaystyle b^3$$
2. $$8\text{;}$$ $$~5\text{;}$$ $$~3\text{;}$$ $$~8-5=3$$

#### 5.

1. $$\displaystyle 10^{15}$$
2. $$15\text{;}$$ $$~3\text{;}$$ $$~15=3\cdot 5$$

#### 7.

1. $$\displaystyle \log_b 2 + \log_b x$$
2. $$\displaystyle \log_b 2 - \log_b x$$

#### 9.

1. $$\displaystyle 1 + 4\log_3 x$$
2. $$\displaystyle \dfrac{1}{t}\log_5 1.1$$

#### 11.

1. $$\displaystyle \dfrac{1}{2} + \dfrac{1}{2}\log_b x$$
2. $$\displaystyle \dfrac{1}{3}\log_3 (x^2+1)$$

#### 13.

1. $$\displaystyle \log P_0 + t\log (1-m)$$
2. $$\displaystyle 4t [\log_4(4+r)-1]$$

#### 15.

1. $$\displaystyle \log_b 4$$
2. $$\displaystyle \log_4(x^2y^3)$$

#### 17.

1. $$\displaystyle \log 2x^{5/2}$$
2. $$\displaystyle \log (t-4)$$

#### 19.

1. $$\displaystyle \log \dfrac{1}{27}$$
2. $$\displaystyle \log_6 (2w^2)$$

#### 21.

1. $$\displaystyle 1.7917$$
2. $$\displaystyle -0.9163$$

#### 23.

1. $$\displaystyle 2.1972$$
2. $$\displaystyle 1.9560$$

#### 25.

$$2.8074$$

#### 27.

$$0.8928$$

#### 29.

$$\pm 1.3977$$

#### 31.

$$-1.6092$$

#### 33.

$$0.2736$$

#### 35.

$$-12.4864$$

#### 37.

1. $$\displaystyle S (t) = S_0(1.09)^t$$
2. $$4.7$$ hours

#### 39.

1. $$\displaystyle C (t) = 0.7(0.80)^t$$
2. After $$2.5$$ hours

#### 41.

1. $$\displaystyle J(t) = 1,041,000\cdot 1.0182^t$$
2. In $$2040$$

#### 43.

1. $$\displaystyle S(t) = S_0 \cdot 0.9527^t$$
2. $$28.61$$ hours

#### 45.

1. $$\displaystyle 5$$
2. $$\displaystyle 6$$
3. $$\displaystyle 5$$
(a) and (c) are equal.

#### 47.

1. $$\displaystyle 6$$
2. $$\displaystyle 9$$
3. $$\displaystyle 6$$
(a) and (c) are equal.

#### 49.

1. $$\displaystyle \log 24\approx 1.38$$
2. $$\displaystyle \log 240\approx 2.38$$
3. $$\displaystyle \log 230\approx 2.36$$
None are equal.

#### 51.

1. $$\displaystyle \log 60\approx 1.78$$
2. $$\displaystyle \log 5\approx 0.70$$
3. $$\displaystyle \dfrac{\log_{10}75}{\log_{10}15}\approx 1.59$$
None are equal.

#### 53.

$$12.9\%$$

#### 55.

About $$11$$ years

#### 57.

1. $$\displaystyle A=1000\left(1+\dfrac{0.12}{n} \right)^{5n}$$
2. $$A$$ increases.
3. $$16\text{;}$$ $$31\text{;}$$ $$553$$
4. Increasing, concave down, asymptotically approaching $$A\approx 1822.12$$

#### 59.

$$k=\dfrac{1}{t}\dfrac{\log(N/N_0)}{\log a}$$

#### 61.

$$t=\dfrac{1}{k}\log\left(\dfrac{A}{A_0}+1\right)$$

#### 63.

$$q=\dfrac{\log(w/p)}{\log v}$$

#### 65.

1. $$x=b^m \text{,}$$ $$y=b^n$$
2. $$\displaystyle \log_b(b^m\cdot b^n)$$
3. $$\displaystyle \log_b(b^m\cdot b^n)=\log_b b^{m+n}$$
4. $$\displaystyle \log_b b^{m+n}=m+n$$
5. $$\displaystyle \log_b b^{m+n}=(\log_b x)+(\log_b y)$$

#### 67.

1. $$\displaystyle x=b^m$$
2. $$\displaystyle \log_b(b^m)^k$$
3. $$\displaystyle \log_b(b^m)^k=\log_b b^{mk}$$
4. $$\displaystyle \log_b b^{mk}=mk$$
5. $$\displaystyle \log_b b^{mk}=(\log_b x)\cdot k$$

### 4.5Exponential ModelsHomework 4.5

#### 1.

$$A(x) = 0.14(50)^{x/3}$$

#### 3.

$$f(x) = \dfrac{65,536}{729}\left(\dfrac{3}{4} \right)^x$$

#### 5.

$$M(x) = 62,500(0.2)^x$$

#### 7.

$$s(x) = \dfrac{1}{135}(9)^x$$

#### 9.

$$y=\dfrac{4}{3}(3)^{x/4}$$

#### 11.

$$y=50(2)^{-x/4}$$

#### 13.

1. $$\displaystyle y= 2.6 -1.3x$$
2. $$\displaystyle y = 2.6(0.5)^x$$

#### 15.

1. $$\displaystyle y= -36-16x$$
2. $$\displaystyle y = \dfrac{12}{5}(5)^{-x/3}$$

#### 17.

1. $$\displaystyle y= 2.5+0.875x$$
2. $$\displaystyle y = 1.5(2)^{x/2}$$

#### 19.

1. $$P = P_0(1.052)^t\text{;}$$ $$t$$ is the number of years since $$1990\text{.}$$
2. $$\dfrac{\log 2}{\log 1.052}\approx 13.7$$ years

#### 21.

1. $$GDP = 1.028^t$$ million pounds
2. $$\dfrac{\log 2}{\log 1.028}\approx 25.1$$ years
3. $$50.2$$ years

#### 23.

1. $$\dfrac{\log 0.5}{\log 0.946}\approx 12.5$$ hours
2. $$25$$ hours

#### 25.

1. $$\dfrac{\log 0.5}{\log 0.844}\approx 4.1$$ hours
2. $$8.2$$ hours

#### 27.

1. $$\displaystyle P = 2000(2)^{t/5}$$
2. $$\displaystyle 14.87\%$$

#### 29.

1. $$\displaystyle D = D_0 \left(\dfrac{1}{2} \right)^{t/18}$$
2. $$\displaystyle 3.78\%$$

#### 31.

1. $$\displaystyle A = A_0 \left(\dfrac{1}{2} \right)^{t/1620 }$$
2. $$\displaystyle 0.043\%$$

#### 33.

1. $$\displaystyle P = P_0 (2)^{t/25 }$$
2. $$\displaystyle 2.81\%$$

#### 35.

1. $$\displaystyle ab^D = 2\cdot ab^0 = 2a$$
2. $$\displaystyle b^D=2$$
3. $$\displaystyle f (t + D) = ab^{t+D} = a\cdot b^t\cdot b^D = ab^t\cdot 2 = 2 f (t)$$
4. For any value of $$t\text{,}$$ after $$D$$ units of time, the new value of $$f$$ is $$2$$ times the old value.

#### 37.

1. $$\displaystyle ab^R = \frac{1}{3} \cdot ab^0 = \frac{1}{3} a$$
2. $$\displaystyle b^R=\frac{1}{3}$$
3. $$\displaystyle g(t + R) = ab^{t+R} = a\cdot b^t\cdot b^R = ab^t\cdot \frac{1}{3} = \frac{1}{3} g(t)$$
4. For any value of $$t\text{,}$$ after $$R$$ units of time, the new value of $$g$$ is $$\frac{1}{3}$$ times the old value.

#### 39.

1. $$\displaystyle A = A_0 \left(\dfrac{1}{2} \right)^{t/5730}$$
2. About $$760$$ years old

#### 41.

1. $$\displaystyle A = A_0 \left(\dfrac{1}{2} \right)^{t/432}$$
2. About $$220$$ years

#### 43.

$$\approx 30$$ years; $$\approx 33$$ years

#### 45.

$$$445.89\text{;}$$$$$376.50$$

#### 47.

1. $$\displaystyle N(t) = 2200(2)^{t/1.5}$$
2. The given model has a smaller growth factor, $$1.356\text{,}$$ than $$2^{1/1.5}\approx 1.59\text{.}$$
3.  Name of chip Year Mooreâ€™slaw $$N(t)$$ Actualnumber Pentium IV $$2000$$ $$2,306,867,200$$ $$20,427,413$$ $$42,000,000$$ Pentium M (Banias) $$2003$$ $$9,227,468,800$$ $$50,932,200$$ $$77,000,000$$ Pentium M (Dothan) $$2004$$ $$14,647,693,680$$ $$69,064,063$$ $$140,000,000$$
4. About $$2.3$$ years

### 4.6Chapter Summary and ReviewChapter 4 Review Problems

#### 1.

1. $$\displaystyle D = 8(1.5)^{t/5}$$
2. $$18\text{;}$$ $$44$$

#### 3.

1. $$\displaystyle M = 100(0.85)^t$$
2. $$52.2$$ mg; $$19.7$$ mg

#### 5.

$$16n^{2x+10}$$

#### 7.

$$\dfrac{1}{m^{x+2}}$$

#### 9.

$$g(t)=16(0.85)^t$$

#### 11.

$$f (x) = 500\left(\dfrac{1}{5}\right)^x$$

#### 13.

$$4.8\%$$ loss

#### 15.

$$6\%$$ loss

#### 17.

1. $$y$$-intercept $$(0, 6)\text{;}$$ asymptote: $$y = 0$$
2. $$\displaystyle [3.472, 10.368]$$

#### 19.

1. $$x$$-intercept $$\left(\frac{\log 3}{\log 2}, 0 \right) \text{;}$$ $$y$$-intercept $$(0, -2)\text{;}$$ asymptote: $$y = -3$$
2. $$\displaystyle [-2.875,5]$$

#### 21.

$$\dfrac{-4}{3}$$

#### 23.

$$-11$$

#### 25.

1. Not (quite) equivalent
2. $$\displaystyle 2^{1/8}\approx 1.090507733\gt 1.0905$$

#### 27.

1. Equivalent
2. $$\displaystyle \left(\dfrac{1}{3} \right)^{x-2} =\left(\dfrac{1}{3} \right)^{x}\cdot \left(\dfrac{1}{3} \right)^{-2} =\left(\dfrac{1}{3} \right)^{x}\cdot 9$$

#### 29.

1. $$\displaystyle y = 4 + 2^{x+1}$$
2. Shift the graph of $$f~~1$$ unit left, $$4$$ units up.

#### 31.

1. $$\displaystyle y = 6 - 3\cdot 2^{x}$$
2. Scale vertically by $$3\text{,}$$ reflect about $$x$$-axis, shift $$6$$ units up.

#### 33.

$$g$$ eventually grows faster.

#### 35.

$$2^{1.5}\approx 2.83\text{;}$$ $$2.25$$

#### 37.

$$M = M_0(2)^{t/10}\text{,}$$ where $$M$$ is the organic content, $$M_0$$ is the organic content at $$0\degree$$C, and $$t$$ is the temperature in $$\degree$$ Celsius.

#### 39.

$$4$$

#### 41.

$$-1$$

#### 43.

$$-3$$

#### 45.

$$\log_{0.3}(x + 1) = -2$$

#### 47.

$$\dfrac{\log 5.1}{1.3}\approx 0.5433$$

#### 49.

$$\dfrac{\log (2.9/3)}{-0.7}\approx 0.21$$

#### 51.

$$\log_b x + \dfrac{1}{3} \log_b y - 2 \log_b z$$

#### 53.

$$\dfrac{4}{3} \log x -\dfrac{1}{3} \log y$$

#### 55.

$$\log\sqrt[3] {\dfrac{x}{y^{2}}}$$

#### 57.

$$\log {\dfrac{1}{8}}$$

#### 59.

$$\dfrac{\log 63}{\log 3}\approx 3.77$$

#### 61.

$$\dfrac{\log 50}{-0.3\log 6}\approx -7.278$$

#### 63.

$$\dfrac{\log(N/N_0)}{k}$$

#### 65.

1. $$\displaystyle 238$$
2. $$\displaystyle 2010$$

#### 67.

1. $$\displaystyle C = 90(1.06)^t$$

#### 81.

1.  Day $$1$$ $$2$$ $$3$$ $$\cdots$$ $$t$$ $$\cdots$$ $$30$$ Wage (cent) $$2$$ $$4$$ $$8$$ $$\cdots$$ $$2^t$$ $$\cdots$$ $$2^{30}$$
2. $$W(t)=2^t$$ cents
3. $$$327.68\text{;}$$$$$10,737,418.24$$

### 5Logarithmic Functions5.1Inverse FunctionsHomework 5.1

#### 1.

1.  $$x$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$f(x)$$ $$0$$ $$1$$ $$-2$$ $$-1$$
 $$y$$ $$0$$ $$1$$ $$-2$$ $$-1$$ $$f^{-1}(y)$$ $$-1$$ $$0$$ $$1$$ $$2$$
2. $$\displaystyle f^{-1}(1)=0$$
3. $$\displaystyle f^{-1}(-1)=2$$

#### 3.

1.  $$x$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$f(x)$$ $$-1$$ $$1$$ $$3$$ $$11$$
 $$y$$ $$-1$$ $$1$$ $$3$$ $$11$$ $$f^{-1}(y)$$ $$-1$$ $$0$$ $$1$$ $$2$$
2. $$\displaystyle f^{1}(1)=0$$
3. $$\displaystyle f^{-1}(3)=1$$

#### 5.

1. $$f (60)\approx 38\text{.}$$ The car that left the $$60$$-foot skid marks was traveling at $$38$$ mph.
2. $$f^{-1} (60) \approx 150\text{.}$$ The car traveling at $$60$$ mph left $$150$$-foot skid marks

#### 7.

1. $$\displaystyle (60~ \text{hours}, 78~ \text{grams})$$
2. $$f^{-1} (90) \approx 19\text{,}$$ so that the vampire batâ€™s weight has dropped to $$90$$ grams about $$19$$ hours after its last meal.

#### 9.

1. $$g (0.05) = 0.28\text{.}$$ At $$5\%$$ interest, $$\1$$ earns $$\0.28$$ interest in $$5$$ years.
2. $$\displaystyle 8.45\%$$
3. $$\displaystyle g^{-1} (I ) = (I + 1)^{1/5} -1$$
4. $$\displaystyle g^{-1} (0.50)\approx 0.0845$$

#### 11.

1. $$f(0.5) \approx 62.9\text{.}$$ At an altitude of $$0.5$$ miles, you can see $$62.9$$ miles to the horizon.
2. $$0.0126$$ mile, or $$66.7$$ feet
3. $$\displaystyle h = f^{-1}(d) =\dfrac{d^2}{7920}$$
4. $$\displaystyle f^{-1} (10)\approx 0.0126$$

#### 13.

1. $$\displaystyle h^{-1} (3)\approx -4$$
2. $$h^{-1} (x)= 5 - x^2\text{;}$$ $$~ h^{-1} (3) = -4$$

#### 15.

1. $$\displaystyle f^{-1} (y) = 3 \sqrt[3]{y} + 2$$
2. $$\displaystyle f^{-1} ( f (4)) = f^{-1} (8) = 4$$
3. $$\displaystyle f ( f^{-1} (-8)) = f (0) = -8$$

#### 17.

$$6$$

#### 19.

$$\dfrac{2}{9}$$

#### 21.

$$4$$

#### 23.

1.  $$x$$ $$0$$ $$6$$ $$y$$ $$300$$ $$1200$$
2.  $$x$$ $$300$$ $$1200$$ $$y$$ $$0$$ $$6$$

#### 25.

1.  $$x$$ $$0$$ $$1$$ $$2$$ $$y$$ $$5$$ $$20$$ $$100$$
2.  $$x$$ $$5$$ $$20$$ $$100$$ $$y$$ $$0$$ $$1$$ $$2$$

#### 27.

1. $$\displaystyle f^{-1}(x)=\dfrac{x+6}{2}$$

#### 29.

1. $$\displaystyle f^{-1}(x)=\sqrt[3]{x-1}$$

#### 31.

1. $$\displaystyle f^{-1}(x)=\dfrac{1}{x}+1$$

#### 33.

1. Domain: $$(-\infty, 4]\text{;}$$ Range: $$[0,\infty)$$
2. $$\displaystyle g^{-1}(x)=4-x^2$$
3. Domain: $$[0,\infty)\text{;}$$ Range: $$(-\infty,4]$$

(a) and (d)

(a)

(a)

(a) and (b)

#### 43.

1. $$f(x)=4+2x \text{;}$$ IV
2. $$f(x)=2-\dfrac{x}{2} \text{;}$$ III
3. $$f(x)=-4-2x \text{;}$$ I
4. $$f(x)=\dfrac{x}{2} \text{;}$$ II

1. III
2. II
3. I

### 5.2Logarithmic FunctionsHomework 5.2

#### 1.

1.  $$x$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$2^x$$ $$\frac{1}{2}$$ $$1$$ $$2$$ $$4$$
 $$x$$ $$\frac{1}{2}$$ $$1$$ $$2$$ $$4$$ $$\log_2 x$$ $$-1$$ $$0$$ $$1$$ $$2$$

#### 3.

1.  $$x$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$\left(\frac{1}{3}\right)^x$$ $$9$$ $$3$$ $$1$$ $$\frac{1}{3}$$
 $$x$$ $$9$$ $$3$$ $$1$$ $$\frac{1}{3}$$ $$\log_{1/3} x$$ $$-2$$ $$-1$$ $$0$$ $$1$$

#### 5.

1. $$\displaystyle x=10,000$$
2. $$\displaystyle x=10^{8}$$

#### 7.

$$0\lt x \lt 0.01$$

#### 9.

1. $$\displaystyle \log 100,322\approx 5.001$$
2. $$\displaystyle \log 693\approx 2.841$$

#### 11.

1. $$\log (-7)$$ is undefined.
2. $$\displaystyle 6 \log 28\approx 8.683$$

#### 13.

1. $$\displaystyle 15.614$$
2. $$\displaystyle 0.419$$

#### 15.

1. $$\displaystyle 81$$
2. $$\displaystyle 4$$
3. Definition of logarithm base $$3$$
4. $$\displaystyle 1.8$$
5. $$\displaystyle a$$

#### 17.

1. $$\displaystyle 2^8$$
2. $$\displaystyle -2$$

#### 19.

1. $$\displaystyle 2k$$
2. $$\displaystyle x^3$$
3. $$\displaystyle \sqrt{x}$$
4. $$\displaystyle 2m$$

#### 21.

1. $$\displaystyle (9,\infty)$$
2. $$\displaystyle f^{-1} (x) = 3^{x-4} + 9$$

#### 23.

1. $$\displaystyle f^{-1}(x)= \log_4 (100 - x) - 2$$
2. $$\displaystyle f^{-1} (f(1)) = f^{-1}(36)=\log_4(64)-2=1$$
3. $$\displaystyle f\left(f^{-1} (84)\right)= f (0) = 100 - 4^2 = 84$$

1. IV
2. I
3. II
4. III

#### 27.

1. The graph resembles a logarithmic function. The (translated) log function is close to the points but appears too steep at first and not steep enough after $$n = 15\text{.}$$ Overall, it is a good fit.
2. $$f$$ grows (more and more slowly) without bound. $$f$$ will eventually exceed $$100$$ per cent, but no one can forget more than $$100\%$$ of what is learned.

#### 29.

1. $$\displaystyle 10^{1.41}\approx 25.704$$
2. $$\displaystyle 10^{-1.69}\approx 0.020417$$
3. $$\displaystyle 10^{0.52}\approx 3.3113$$

#### 31.

$$16^w = 256$$

#### 33.

$$b^{-2} = 9$$

#### 35.

$$10^{-2.3} = A$$

#### 37.

$$u^{w} = v$$

#### 39.

$$b=2$$

#### 41.

$$b=100$$

#### 43.

$$x=11$$

#### 45.

$$x=7^{2/3}$$

#### 47.

$$x=4$$

#### 49.

$$x=11$$

#### 51.

$$x=3$$

No solution

#### 55.

$$A=k(10^{t/T}-1)$$

#### 57.

$$s=\dfrac{b^{N/N_0}}{k}$$

#### 59.

$$H=(H_0)^{kM^2}$$

1. II
2. VI
3. III
4. V
5. I
6. IV

#### 63.

1. No inverse function
2. No inverse function

#### 65.

The functions are equal.

#### 67.

The functions are equal.

#### 69.

1.  $$x$$ $$x^2$$ $$\log_{10}x$$ $$\log_{10}x^2$$ $$1$$ $$1$$ $$0$$ $$0$$ $$2$$ $$4$$ $$0.301$$ $$0.602$$ $$3$$ $$9$$ $$0.477$$ $$0.954$$ $$4$$ $$16$$ $$0.602$$ $$1.204$$ $$5$$ $$25$$ $$0.699$$ $$1.398$$ $$6$$ $$36$$ $$0.778$$ $$1.556$$
2. $$\displaystyle \log_{10}x^2=2\log_{10}x$$

#### 71.

 $$x$$ $$y=\log_e x$$ $$1$$ $$0$$ $$2$$ $$0.693$$ $$4$$ $$1.386$$ $$16$$ $$2.772$$ $$\frac{1}{2}$$ $$-0.693$$ $$\frac{1}{4}$$ $$-1.386$$ $$\frac{1}{16}$$ $$-2.772$$

### 5.3The Natural BaseHomework 5.3

#### 1.

 $$x$$ $$-10$$ $$-5$$ $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$f(x)$$ $$0.135$$ $$0.368$$ $$1$$ $$2.718$$ $$7.389$$ $$20.086$$ $$54.598$$

#### 3.

 $$x$$ $$-10$$ $$-5$$ $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$f(x)$$ $$20.086$$ $$4.482$$ $$1$$ $$0.223$$ $$0.05$$ $$0.011$$ $$0.00248$$

#### 5.

1. $$\displaystyle 2$$
2. $$\displaystyle 5t$$
3. $$\displaystyle \dfrac{1}{x}$$
4. $$\displaystyle \dfrac{1}{2}$$

#### 7.

1. $$\displaystyle 0.64$$
2. $$\displaystyle 3.81$$
3. $$\displaystyle -1.20$$

#### 9.

1. $$\displaystyle 4.14$$
2. $$\displaystyle 1.88$$
3. $$\displaystyle 0.07$$

#### 11.

1. $$\displaystyle N(t)=6000e^{0.04t}$$
2.  $$t$$ $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$25$$ $$30$$ $$N(t)$$ $$6000$$ $$7328$$ $$8951$$ $$10,933$$ $$13,353$$ $$16,310$$ $$19,921$$
3. $$\displaystyle 15,670$$
4. $$70.3$$ hrs

#### 13.

1. $$941.8$$ lumens
2. $$2.2$$ cm

#### 15.

$$P (t) = 20\left(e^{0.4} \right)^t \approx 20\cdot 1.492^t\text{;}$$ increasing; initial value $$20$$

#### 17.

$$P (t) = 6500\left(e^{-2.5} \right)^t \approx 6500\cdot 0.082^t\text{;}$$ decreasing; initial value $$6500$$

#### 19.

1.  $$x$$ $$0$$ $$0.5$$ $$1$$ $$1.5$$ $$2$$ $$2.5$$ $$e^x$$ $$1$$ $$1.6487$$ $$2.7183$$ $$4.4817$$ $$7.3891$$ $$12.1825$$
2. Each ratio is $$e^{0.5} \approx 1.6487\text{:}$$ Increasing $$x$$-values by a constant $$\Delta x = 0.5$$ corresponds to multiplying the $$y$$-values of the exponential function by a constant factor of $$e^{\Delta x}\text{.}$$

#### 21.

1.  $$x$$ $$0$$ $$0.6931$$ $$1.3863$$ $$2.0794$$ $$2.7726$$ $$3.4657$$ $$4.1589$$ $$e^x$$ $$1$$ $$2$$ $$4$$ $$8$$ $$16$$ $$32$$ $$64$$
2. Each difference in $$x$$-values is approximately $$\ln 2\approx 0.6931\text{:}$$ Increasing $$x$$-values by a constant $$\Delta x = \ln 2$$ corresponds to multiplying the $$y$$-values of the exponential function by a constant factor of $$e^{\Delta x} = e^{\ln 2} = 2\text{.}$$ That is, each function value is approximately equal to double the previous one.

#### 23.

$$0.8277$$

#### 25.

$$-2.9720$$

#### 27.

$$1.6451$$

#### 29.

$$-3.0713$$

#### 31.

$$t=\dfrac{1}{k}\ln y$$

#### 33.

$$t=\ln \left(\dfrac{k}{k-y}\right)$$

#### 35.

$$k=e^{T/T_0}-10$$

#### 37.

1.  $$n$$ $$0.39$$ $$3.9$$ $$39$$ $$390$$ $$\ln n$$ $$-0.942$$ $$1.361$$ $$3.664$$ $$5.966$$
2. Each difference in function values is approximately $$\ln 10\approx 2.303\text{:}$$ Multiplying $$x$$-values by a constant factor of $$10$$ corresponds to adding a constant value of ln 10 to the $$y$$-values of the natural log function.

#### 39.

1.  $$n$$ $$2$$ $$4$$ $$8$$ $$16$$ $$\ln n$$ $$0.693$$ $$1.386$$ $$2.079$$ $$2.773$$
2. Each quotient equals $$k\text{,}$$ where $$n = 2^k\text{.}$$ Because $$\ln n = \ln 2^k = k\cdot \ln 2\text{,}$$ $$k = \dfrac{\ln n}{\ln 2}\text{.}$$

#### 41.

1. $$\displaystyle N (t) = 100e^{(\ln 2)t}\approx 100e^{0.6931t}$$

#### 43.

1. $$\displaystyle N (t) = 1200e^{(\ln 0.6)t}\approx 1200e^{-0.5108t}$$

#### 45.

1. $$\displaystyle N (t) = 10e^{(\ln 1.15)t}\approx 10e^{0.1398t}$$

#### 47.

1. $$\displaystyle 20,000$$
2. $$\displaystyle \left(\dfrac{35,000}{20,000} \right)^{1/10}\approx e^{0.056}$$
3. $$\displaystyle P(t) = 20,000e^{0.056t}$$
4. $$\displaystyle 107,188$$

#### 49.

1. $$\displaystyle \left(\dfrac{385}{500} \right)^{1/2}\approx e^{-0.1307}$$
2. $$\displaystyle N(t) = 500e^{-0.1307t}$$
3. $$135.3$$ mg

#### 51.

1. $$\displaystyle A(t) = 500e^{0.095t}$$
2. $$7.3$$ years
3. $$7.3$$ years
dâ€“e

#### 53.

1. $$6$$ hours
2. $$6$$ hours

#### 55.

1. $$\frac{1}{2}N_0 \text{,}$$ $$\frac{1}{4}N_0 \text{,}$$ $$\frac{1}{16}N_0$$
2. $$\displaystyle N (t) = N_0e^{-0.0866t}$$

#### 57.

1. $$\displaystyle y = 116 (0.975)^t$$
2. $$\displaystyle G (t) = 116e^{-0.025t}$$
3. $$28$$ minutes

### 5.4Logarithmic ScalesHomework 5.4

#### 5.

$$1.58\text{,}$$ $$6.31\text{,}$$ $$15.8\text{,}$$ $$63.1$$

#### 7.

$$1\text{,}$$ $$80\text{,}$$ $$330\text{,}$$ $$1600\text{,}$$ $$7000\text{,}$$ $$4\times 10^7$$

#### 11.

Proxima Centauri: $$15.5\text{;}$$ Barnard: $$13.2\text{;}$$ Sirius: $$1.4\text{;}$$ Vega: $$0.6\text{;}$$ Arcturus: $$-0.4\text{;}$$ Antares: $$-4.7\text{;}$$ Betelgeuse: $$-7.2$$

#### 13.

1. $$\displaystyle 1$$
2. $$\displaystyle 0.5012$$
3. $$\displaystyle 0.1259$$
4. $$\displaystyle 0.01$$
5. $$\displaystyle 0.000079$$
6. $$\displaystyle 3.2\times 10^{-7}$$
7. $$\displaystyle 2\times 10^{-8}$$
8. $$\displaystyle 8\times 10^{-10}$$

#### 15.

1. $$\displaystyle 10^{1.75}\approx 56.2341$$
2. $$\displaystyle 10^{(\log 600)/2}\approx 24.4949$$

#### 17.

$$10^{3.4} \approx 2512$$

#### 19.

A: $$a\approx 45\text{,}$$ $$p \approx 7.4\%\text{;}$$ B: $$a \approx 400\text{,}$$ $$p \approx 15\%\text{;}$$ C: $$a\approx 6000\text{,}$$ $$p\approx 50\%\text{;}$$ D: $$a \approx 13000\text{,}$$ $$p \approx 45\%$$

#### 21.

$$3.2$$

#### 23.

$$0.0126$$

#### 25.

$$100$$

#### 27.

$$6,309,573$$ watts per square meter

#### 29.

$$1000$$

#### 31.

$$12.6$$

#### 33.

$$100$$

#### 35.

$$\approx 25,000$$

#### 37.

$$4.7$$

#### 39.

$$53$$

### 5.5Chapter Summary and ReviewChapter 5 Review Problems

#### 1.

 $$y$$ $$-1$$ $$1$$ $$3$$ $$11$$ $$x=f^{-1}(y)$$ $$-1$$ $$0$$ $$1$$ $$2$$

#### 3.

 $$y$$ $$0$$ $$\frac{-1}{3}$$ $$-1$$ $$-3$$ $$w=g^{-1}(y)$$ $$-1$$ $$0$$ $$1$$ $$2$$

#### 5.

1. $$\displaystyle P^{-1}(350)=40$$
2. $$\displaystyle P^{-1}(100)=0$$

#### 7.

1. $$\displaystyle f^{-1} (x) = x - 4$$

#### 9.

1. $$\displaystyle f^{-1} (x) =\sqrt[3]{x+1}$$

#### 11.

1. $$\displaystyle f^{-1} (x) =\dfrac{1}{x-2}$$

#### 13.

$$0$$

#### 15.

1. $$f^{-1} (300) = 200\text{:}$$ $$\200,000$$ in advertising results in $$\300,000$$ in revenue.
2. $$f (A) = 250$$ or $$A = f^{-1} (250)$$

#### 17.

$$10^z = 0.001$$

#### 19.

$$2^{x-2} = 3$$

#### 21.

$$b^{3} = 3x+1$$

#### 23.

$$n^{p-1} = q$$

#### 25.

$$6n$$

#### 27.

$$2x+6$$

#### 29.

$$-1$$

#### 31.

$$\dfrac{1}{2}$$

#### 33.

$$4$$

#### 35.

$$\dfrac{-15}{8}$$

#### 37.

$$\dfrac{9}{4}$$

#### 39.

$$3$$

#### 41.

$$x\approx 1.548$$

#### 43.

$$x\approx 411.58$$

#### 45.

$$x\approx 2.286$$

#### 47.

$$\sqrt{x}$$

#### 49.

$$k-3$$

#### 51.

1. $$\displaystyle P = 7,894,862e^{-0.011t}$$
2. $$\displaystyle 1.095\%$$

#### 53.

1. \$$$1419.07$$
2. $$13.9$$ years
3. $$\displaystyle t = 20 \ln\left(\dfrac{A}{1000} \right)$$

#### 55.

$$t=\dfrac{-1}{k}\ln\left(\dfrac{y-6}{12} \right)$$

#### 57.

$$M=N^{Qt}$$

#### 59.

$$P (t) = 750 (1.3771)^t$$

#### 61.

$$N(t) = 600 e^{-0.9163t}$$

#### 65.

Order $$3\text{:}$$ $$17,000\text{;}$$ Order $$4\text{:}$$ $$5000\text{;}$$ Order $$8\text{:}$$ $$40\text{;}$$ Order $$9\text{:}$$ $$11$$

#### 67.

$$5\times 10^{-7}$$

#### 69.

$$3160$$

### 6Quadratic Functions6.1Factors and $$x$$-InterceptsHomework 6.1

#### 1.

1.  $$t$$ $$0$$ $$0.5$$ $$1$$ $$1.5$$ $$2$$ $$2.5$$ $$3$$ $$3.5$$ $$4$$ $$4.5$$ $$5$$ $$h$$ $$300$$ $$306$$ $$304$$ $$294$$ $$276$$ $$250$$ $$216$$ $$174$$ $$124$$ $$66$$ $$0$$
2. $$306.25$$ ft at $$0.625$$ sec
3. $$1.25$$ sec
4. $$5$$ sec

#### 3.

$$\dfrac{-5}{2} \text{,}$$ $$~2$$

#### 5.

$$0 \text{,}$$ $$~\dfrac{-10}{3}$$

#### 7.

$$\dfrac{-3}{4} \text{,}$$ $$~-8$$

#### 9.

$$4$$

#### 11.

$$\dfrac{1}{2} \text{,}$$ $$~-3$$

#### 13.

$$0 \text{,}$$ $$~3$$

#### 15.

$$1$$

#### 17.

$$\dfrac{1}{2} \text{,}$$ $$~1$$

#### 19.

$$2 \text{,}$$ $$~3$$

#### 21.

$$-1 \text{,}$$ $$~2$$

#### 23.

$$-3 \text{,}$$ $$~6$$

#### 25.

The 3 graphs have the same $$x$$-intercepts. In general, the graph of $$y = ax^2 + bx + c$$ has the same $$x$$-intercepts as the graph of $$y = k(ax^2 + bx + c)\text{.}$$

#### 27.

The 3 graphs have the same $$x$$-intercepts. In general, the graph of $$y = ax^2 + bx + c$$ has the same $$x$$-intercepts as the graph of $$y = k(ax^2 + bx + c)\text{.}$$

#### 29.

$$x^2 + x - 2 = 0$$

#### 31.

$$x^2 + 5x = 0$$

#### 33.

$$2x^2 + 5x - 3 = 0$$

#### 35.

$$8x^2 -10x -3 = 0$$

#### 37.

$$f(x) = 0.1(x - 18)(x + 15)$$

#### 39.

$$g(x) = -0.08(x - 18)(x + 32)$$

#### 41.

1. $$\displaystyle 10^2 + h^2 = (h + 2)^2$$
2. $$24$$ ft

#### 43.

1. $$\displaystyle h=-16t^2 + 16t + 8$$
2. $$12$$ ft; $$8$$ ft
3. $$11=-16t^2+16t+8\text{;}$$ at $$\dfrac{1}{4}$$ sec and $$\dfrac{3}{4}$$ sec
4. $$\displaystyle \Delta \text{Tbl}=0.25$$
5. $$1.37$$ sec

#### 45.

1.  Width Length Area $$10$$ $$170$$ $$1700$$ $$20$$ $$160$$ $$3200$$ $$30$$ $$150$$ $$4500$$ $$40$$ $$140$$ $$5600$$ $$50$$ $$130$$ $$6500$$ $$60$$ $$120$$ $$7200$$ $$70$$ $$110$$ $$7700$$ $$80$$ $$100$$ $$8000$$
2. $$l= 180 - x\text{,}$$ $$A = 180x - x^2\text{;}$$ $$80$$ yd by $$100$$ yd
3. $$180x-x^2=8000\text{,}$$ $$80$$ yd by $$100$$ yd, or $$100$$ yd by $$80$$ yd. There are two solutions because the pasture can be oriented in two directions.

#### 47.

1. $$l = x - 4\text{,}$$ $$~w = x - 4\text{,}$$ $$~h = 2\text{,}$$ $$~V = 2(x - 4)^2$$
2.  $$x$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$V$$ $$0$$ $$2$$ $$8$$ $$18$$ $$32$$ $$50$$ $$72$$
3. As $$x$$ increases, $$V$$ increases.
4. $$9$$ inches by $$9$$ inches.
5. $$2(x - 4)^2 = 50\text{,}$$ $$~x = 9$$

#### 49.

1.  $$x$$ $$0$$ $$500$$ $$1000$$ $$1500$$ $$2000$$ $$2500$$ $$3000$$ $$3500$$ $$I$$ $$0$$ $$550$$ $$1000$$ $$1350$$ $$1600$$ $$1750$$ $$1800$$ $$1750$$
 $$x$$ $$4000$$ $$4500$$ $$5000$$ $$5500$$ $$6000$$ $$6500$$ $$7000$$ $$I$$ $$1600$$ $$1350$$ $$1000$$ $$550$$ $$0$$ $$-650$$ $$-1400$$
2. $$1600\text{,}$$ $$1000\text{,}$$ $$-1400$$
3. No increase
4. $$3000\text{;}$$ $$1800$$

#### 51.

$$\pm 1$$

#### 53.

$$\sqrt[3]{-3/4} \text{,}$$ $$1$$

#### 55.

$$-27 \text{,}$$ $$1$$

#### 57.

$$\log 2 \text{,}$$ $$\log 3$$

#### 59.

$$1 \text{,}$$ $$2$$

#### 61.

$$\dfrac{-1}{6} \text{,}$$ $$1$$

#### 63.

1. $$\displaystyle A=\dfrac{1}{2}(x^2-y^2)$$
2. $$\displaystyle A=\dfrac{1}{2}(x-y)(x+y)$$
3. $$18$$ sq ft

#### 1.

1. $$\displaystyle (x+4)^2$$
2. $$\displaystyle \left(x-\dfrac{7}{2} \right)^2$$
3. $$\displaystyle \left(x+\dfrac{3}{4} \right)^2$$
4. $$\displaystyle \left(x-\dfrac{2}{5} \right)^2$$

#### 3.

$$1$$

#### 5.

$$-4\text{,}$$ $$~-5$$

#### 7.

$$\dfrac{3}{2} \pm \sqrt{\dfrac{21}{4}} = \dfrac{-3\pm\sqrt{21}}{2}$$

#### 9.

$$-1\pm \sqrt{\dfrac{5}{2}}$$

#### 11.

$$\dfrac{-4}{3} \text{,}$$ $$~1$$

#### 13.

$$\dfrac{1}{4} \pm \sqrt{\dfrac{13}{16}} = \dfrac{1\pm\sqrt{13}}{4}$$

#### 15.

$$-1 \text{,}$$ $$\dfrac{4}{3}$$

#### 17.

$$-2 \text{,}$$ $$\dfrac{2}{5}$$
$$-1\pm\sqrt{1-c}$$
$$-\dfrac{b}{2} \pm \sqrt{\dfrac{b^2-4}{4}} = \dfrac{-b\pm\sqrt{b^2-4}}{2}$$