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Section 2.4 Functions as Mathematical Models

Subsection 1. Recognize familiar formulas

You can review the formulas for volume in the Toolkit for Section 2.1. Some other useful formulas appear below.

Subsubsection Examples

Example 2.42.

Write a formula for the volume of a rectangular box, and identify the variables.

Solution

\(V=lwh~~\)

\(V\) stands for volume, and \(l,~w,\) and \(h\) stand for, respectively, the length, width, and height of the box.

Example 2.43.

Write a formula for the average of a number of scores, and identify the variables.

Solution

\(A=\dfrac{S}{n}~~\)

\(A\) stands for the average, \(S\) stands for the sum of the scores, and \(n\) stands for the number of scores.

Subsubsection Exercises

Choose the correct formula from the list below, and identify the variables.

  • \(\displaystyle I=Prt\)
  • \(\displaystyle P=R-C\)
  • \(\displaystyle d=rt\)
  • \(\displaystyle P=rW\)
  1. The distance traveled at a constant speed.
  2. The simple interest on an investment.
  3. The part specified by a percentage.
  4. The profit on sales of an item.
Answer

  1. \(d=rt\)

    \(~~d\) stands for the distance traveled at speed \(r\) for time \(t\text{.}\)

  2. \(I=Prt\)

    \(~~I\) stands for the interest earned on an investment \(P\) at interest rate \(r\) after a time period \(t\text{.}\)

  3. \(P=rW\)

    \(~~P\) stands for the quantity \(r\) percent of a whole amount \(W\text{.}\)

  4. \(P=R-C\)

    \(~~P\) stands for the profit left after the costs \(C\) are subtracted from the revenue \(R\text{.}\)

Choose the correct geometric formula from the list below, and identify the variables.

  • \(\displaystyle A=lw\)
  • \(\displaystyle P=2l+2w\)
  • \(\displaystyle A=\pi r^2\)
  • \(\displaystyle C=\pi d\)
  • \(\displaystyle V=\pi r^2h\)
  • \(\displaystyle V=\dfrac{4}{3}\pi r^3\)
  1. The volume of a cylinder.
  2. The area of a circle.
  3. The area of a rectangle.
  4. The perimeter of a rectangle.
  5. The volume of a sphere.
  6. The circumference of a circle.
Answer

  1. \(V=\pi r^2h\)

    \(~~V\) stands for the volume, \(r\) for the radius, and \(h\) for the height of the cylinder.

  2. \(A=\pi r^2\)

    \(~~A\) stands for the area and \(r\) for the radius of the circle.

  3. \(A=lw\)

    \(~~A\) stands for the area of the rectangle, \(l\) and \(w\) stand for its length and width.

  4. \(P=2l+2w\)

    \(~~P\) stands for the perimeter of the rectangle, \(l\) and \(w\) stand for its length and width.

  5. \(V=\dfrac{4}{3}\pi r^3\)

    \(~~V\) stands for the volume and \(r\) for the radius of the sphere.

  6. \(C=\pi d\)

    \(~~C\) stands for the circumference and \(d\) for the diameter of the circle.

Subsection 2. Calculate slope between points

Recall the two-point formula for slope:

\begin{equation*} m = \dfrac{y_2-y_1}{x_2-x_1} \end{equation*}

If the slope of a graph is increasing, we say that the graph is concave up; if the slopes are decreasing, the graph is concave down.

Subsubsection Example

Example 2.46.

Here is a table of values for a function \(f(x)\text{.}\)

\(x\) \(1\) \(2\) \(5\) \(9\) \(15\)
\(f(x)\) \(10\) \(14\) \(22\) \(30\) \(39\)

  1. Calculate the slopes of the line segments joining each successive pair of points on the graph.
  2. According to your calculations, is the graph concave up or down?
Solution
  1. \(m_1=\dfrac{14-10}{2-1}=4\text{,}\) \(m_2=\dfrac{22-14}{5-2}=\dfrac{8}{3}\text{,}\) \(m_3=\dfrac{30-22}{9-5}=2\text{,}\) \(m_4=\dfrac{39-30}{15-9}=\dfrac{3}{2}\)
  2. The slopes are decreasing, so the graph is concave down.

Subsubsection Exercises

Here is a table of values for a function \(h(t)\text{.}\)

\(t\) \(-2\) \(0\) \(1\) \(4\) \(5\)
\(h(t)\) \(-10\) \(-2\) \(0.5\) \(4\) \(4.5\)

  1. Calculate the slopes of the line segments joining each successive pair of points on the graph.
  2. According to your calculations, is the graph concave up or down?
Answer
  1. \(\displaystyle m_1=4,~~m_2=2.5,~~m_3=\dfrac{7}{6},~~m_4=0.5\)
  2. Concave down.
  1. Complete the table for the function \(g(x)=\dfrac{16}{x}\text{.}\)
    \(x\) \(1\) \(2\) \(4\) \(8\) \(16\)
    \(g(x)\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\)
  2. Calculate the slopes of the line segments joining each successive pair of points on the graph.
  3. According to your calculations, is the graph concave up or down?
Answer
  1. \(x\) \(1\) \(2\) \(4\) \(8\) \(16\)
    \(g(x)\) \(16\) \(8\) \(4\) \(2\) \(1\)
  2. \(\displaystyle m_1=-8,~~m_2=-2,~~m_3=\dfrac{-1}{2},~~m_4=\dfrac{-1}{8}\)
  3. Concave up.

Subsection 3. Evaluate a piecewise function

In a "piecewise" function, the \(x\)-axis is divided into several pieces or regions, and the function has a different formula on each piece.

Subsubsection Example

Example 2.49.

Evaluate the function.

\begin{equation*} f(x)=\begin{cases} 4 \amp x \lt 1\\ 5+2x-x^2 \amp 1\le x\lt 3\\ 2x \amp x \gt 3 \end{cases} \end{equation*}
graph
  1. \(\displaystyle f(-2)\)
  2. \(\displaystyle f(1)\)
  3. \(\displaystyle f(2)\)
  4. \(\displaystyle f(3)\)
Solution
  1. Because \(-2\) lies in the first region, \(f(-2)=4\)
  2. \(1\) lies in the second region, so \(f(\alert{1})=5+2(\alert{1})-(\alert{1})^2 = 6\)
  3. \(2\) lies in the second region, so \(f(\alert{2}) = 5+2(\alert{2})-(\alert{2})^2 = 5\)
  4. \(3\) does not lie in either the second or third region, so \(f(3)\) is undefined. Notice that there are open dots on the graph at \(x=3\text{.}\)

Subsubsection Exercise

Sketch a graph of the function, and evaluate for the inputs below.

\begin{equation*} g(x)=\begin{cases} x+2 \amp x \lt -1\\ 1 \amp -1\lt x\lt 1\\ x \amp x \ge 1 \end{cases} \end{equation*}
grid
  1. \(\displaystyle g(-2)\)
  2. \(\displaystyle g(-1)\)
  3. \(\displaystyle g(0)\)
  4. \(\displaystyle g(1)\)
Answer
graph
  1. \(\displaystyle 0\)
  2. undefined
  3. \(\displaystyle 1\)
  4. \(\displaystyle 1\)