Intermediate Algebra: Functions and Graphs

1.

Combining powers and roots

1. Start with 8, square it, and then take the cube root of the result. That is:

   \begin{equation*} \sqrt[3]{8^2} = \sqrt[3]{?}= \end{equation*}

   Now start with 8, take the cube root, then square the result. That is:

   \begin{equation*} (\sqrt[3]{8})^2 = (?)^2= \end{equation*}

2. Do you get the same answers for parts (a) and (b)? You should! Now try the same thing with some irrational numbers. Use your calculator, and round your answers to two decimal places.

   \begin{align*} \sqrt{3^5} \amp = \amp \amp (\sqrt{3})^5 = \\ (\sqrt[3]{58})^4\amp = \amp \amp \sqrt[3]{58^4}= \end{align*}

3. Choose the most convenient order of operations to evaluate each root without using a calculator.

   1. \(\displaystyle \sqrt[5]{32^3}\)
   2. \(\displaystyle -\sqrt[3]{27^4}\)

2.

Using exponents

1. Simplify \((\sqrt[3]{8})^2\) using exponents as follows. Fill in the blanks with the correct exponents.

   \begin{equation*} (\sqrt[3]{8})^2 = (8^{\underline{\hspace{0.909090909090909em}}})^2 = 8^{\underline{\hspace{0.909090909090909em}}} \end{equation*}
2. Use your calculator to compute \(8^{2/3}\text{.}\) Don't forget to put parentheses around the exponent.

3. Approximate each power to the nearest thousandth.

   1. \(\displaystyle 12^{5/6}\)
   2. \(\displaystyle \sqrt[3]{6^4}\)
   3. \(\displaystyle 37^{-2/3}\)
   4. \(\displaystyle 4.7^{2.3}\)

3.

Evaluate each root without using a calculator.

1. \(\displaystyle \sqrt[4]{16y^{12}} \)

2. \(\displaystyle \sqrt[5]{243x^{10}}\)

3. \(\displaystyle \sqrt[3]{8x^9y^{27}} \)

4. \(\displaystyle -\sqrt[4]{81a^8b^{12}}\)

5. \(\displaystyle \sqrt[3]{64x^6 y^{18}}\)

6. \(\displaystyle -\sqrt[5]{32x^{25}y^{5}} \)

4.

Evaluate each power without using a calculator.

1. \(\displaystyle 81^{3/4} \)

2. \(\displaystyle 125^{2/3} \)

3. \(\displaystyle 625^{0.75} \)

4. \(\displaystyle 16^{-3/2} \)

5. \(\displaystyle 8^{-4/3} \)

6. \(\displaystyle 32^{-1.6} \)

21.

If \(f (x) = (3x - 4)^{3/2}\text{,}\) find \(x\) so that \(f (x) = 27\text{.}\)

22.

If \(S(x) = 12x^{-5/4}\text{,}\) find \(x\) so that \(S(x) = 20\text{.}\)

31.

1. Complete the table of values, then graph both functions on the grid, along with the graph of \(y=x\text{.}\)

   \begin{equation*} f(x) = x^{4/3},\quad g(x)=x^{3/2} \end{equation*}

   \(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)
   \(f(x)\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)
   \(g(x)\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)

   

2. Check your work by graphing the functions in the suggested window, and compare the graphs.

   \begin{align*} \text{Xmin} \amp = 0 \amp\amp \text{Xmax} = 6\\ \text{Ymin} \amp = 0 \amp\amp \text{Ymax} = 10 \end{align*}

32.

1. Complete the table of values, then graph both functions on the grid, along with the graph of \(y=x\text{.}\)

   \begin{equation*} f(x) = x^{3/4}, \quad g(x)=x^{2/3} \end{equation*}

   \(x\)	\(0\)	\(2\)	\(4\)	\(6\)	\(8\)	\(10\)
   \(f(x)\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)
   \(g(x)\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)	\(~~~\)

   

2. Check your work by graphing the functions in the suggested window, and compare the graphs.

   \begin{align*} \text{Xmin} \amp = 0 \amp\amp \text{Xmax} = 10\\ \text{Ymin} \amp = 0 \amp\amp \text{Ymax} = 4 \end{align*}

35.

During a flu epidemic in a small town, health officials estimate that the number of people infected \(t\) days after the first case was discovered is given by

1. Complete the table of values.

   \(t\)	\(5\)	\(10\)	\(15\)	\(20\)
   \(I(t)\)	\(\)	\(\)	\(\)	\(\)

2. How long will it be before 300 people are ill?
3. Graph the function \(I(t)\) on your calculator, and verify your answer to part (b) on your graph.

36.

The research division of an advertising firm estimates that the number of people who have seen their ads \(t\) days after the campaign begins is given by the function

1. Complete the table of values.

   \(t\)	\(6\)	\(10\)	\(14\)	\(20\)
   \(N(t)\)	\(\)	\(\)	\(\)	\(\)

2. How long will it be before 75,000 people have seen the ads?
3. Graph the function \(n(t)\) on your calculator, and verify your answer to part (b) on your graph.

37.

In the 1970s, Jared Diamond studied the number of bird species on small islands near New Guinea. He found that larger islands support a larger number of different species, according to the formula

where \(S\) is the number of species on an island of area \(A\) square kilometers. (Source: Chapman and Reiss, 1992)

1. Fill in the table.

   \(A\)	\(10\)	\(100\)	\(1000\)	\(5000\)	\(10,000\)
   \(S\)	\(\hphantom{000000} \)	\(\hphantom{000000} \)	\(\hphantom{000000} \)	\(\hphantom{000000} \)	\(\hphantom{000000} \)

2. Graph the function on your calculator for \(0\lt A\le 10,000\text{.}\)

3. How many species of birds would you expect to find on Manus Island, with an area of 2100 square kilometers? On Lavongai, which bird's area is 1140 square kilometers?

4. How large must an island be in order to support 200 different species of bird?

38.

The climate of a region has a great influence on the types of animals that can survive there. Extreme temperatures create difficult living conditions, so the diversity of wildlife decreases as the annual temperature range increases. Along the west coast of North America, the number of species of mammals, \(M\text{,}\) is approximately related to the temperature range, \(R\text{,}\) (in degrees Celsius) by the function

(Source: Chapman and Reiss, 1992)

1. Graph the function for temperature ranges up to \(30\degree\)C.

2. How many species would you expect to find in a region where the temperature range is \(10\degree \)C? Label the corresponding point on your graph.

3. If 50 different species are found in a certain region, what temperature range would you expect the region to experience? Label the corresponding point on your graph.

4. Evaluate the function to find \(f(9)\text{,}\) \(f(10)\text{,}\) \(f(19)\text{,}\) and \(f(20)\text{.}\) What do these values represent? Calculate the change in the number of species as the temperature range increases from \(9\degree\)C to \(10\degree\)C and from \(19\degree\)C to \(20\degree\)C. Which \(1\degree\) increase results in a greater decrease in diversity? Explain your answer in terms of slopes on your graph.

39.

1. The average body mass of a dolphin is about 140 kilograms, twice the body mass of an average human male. Using the allometric equation above, calculate the ratio of the brain mass of a dolphin to that of a human.

2. A good-sized brown bear weighs about 280 kilograms, twice the weight of a dolphin. Calculate the ratio of the brain mass of a brown bear to that of a dolphin.

3. Use a ratio to compare the heartbeat frequencies of a dolphin and a human, and those of a brown bear and a dolphin.

40.

A bicycle ergometer is used to measure the amount of power generated by a cyclist. The scatterplot shows how long an athlete was able to sustain variouslevels of power output. The curve is the graph of

which approximately models the data. (Source: Alexander, 1992)

1. As represented by the graph, which variable is the input, and which is the output?
2. The athlete maintained 650 watts of power for 40 seconds. What power output does the equation predict for 40 seconds?
3. The athlete maintained 300 watts of power for 10 minutes. How long does the equation predict that power output can be maintained?
4. In 1979 a remarkable pedal-powered aircraft called the Gossamer Albatross was successfully flown across the English Channel. The flight took 3 hours. According to the equation, what level of power can be maintained for 3 hours?
5. The Gossamer Albatross needed 250 watts of power to keep it airborne. For how long can 250 watts be maintained, according to the equation?

41.

Birds’ eggs typically lose 10%–20% of their mass during incubation. The embryo metabolizes lipid during growth, and this process releases water vapor through the porous shell. The incubation time for birds’ eggs is a function of the mass of the egg and has been experimentally determined as

where \(m\) is measured in grams and \(I\) in days. (Source: Burton, 1998)

1. Calculate the incubation time (to the nearest day) for the wren, whose eggs weigh about 2.5 grams, and the greylag goose, whose eggs weigh 46 grams.
2. The rate of water loss from the egg is also a function of its mass and appears to follow the rule
   \begin{equation*} W(m) = 0.015 m^{0.742} \end{equation*}
   Combine the functions \(I(m)\) and \(W(m)\) to calculate the fraction of the initial egg mass that is lost during the entire incubation period.
3. Explain why your result shows that most eggs lose about 18% of their mass during incubation.

42.

The incubation time for birds’ eggs is given by

where \(m\) is measured in grams and \(I\) in days. (See Problem 41.) Before hatching, the eggs take in oxygen at the rate of

in milliliters per day. (Source: Burton, 1998)

1. Combine the functions \(I(m)\) and \(O(m)\) to calculate the total amount of oxygen taken in by the egg during its incubation.
2. Use your result from part (a) to explain why total oxygen consumption per unit mass is approximately inversely proportional to incubation time. (Oxygen consumption is a reliable indicator of metabolic rate, and it is reasonable that incubation time should be inversely proportional to metabolic rate.)
3. Predict the oxygen consumption per gram of a herring gull’s eggs, given that their incubation time is 26 days. (The actual value is 11 milliliters per day.)

43.

Kepler’s law gives a relation between the period, \(p\text{,}\) of a planet’s revolution and its average distance, \(a\text{,}\) from the Sun:

where \(K = 1.243 \times 10^{24}\text{,}\) \(a\) is measured in miles, and \(p\) is measured in years.

1. Solve Kepler’s law for \(p\) in terms of \(a\text{.}\)
2. Find the period of Mars if its average distance from the Sun is \(1.417 \times 10^8\) miles.

44.

Refer to Kepler's law, \(p^2=Ka^3\text{,}\) in Problem 43.

1. Solve Kepler’s law for \(a\) in terms of \(p\text{.}\)
2. The period of Venus is 0.615 years. Find the distance from Venus to the Sun.