Chapter 3

Assume that \(f\) is a one-to-one function. $$ \begin{array}{l}{\text { (a) If } f(2)=7, \text { find } f^{-1}(7)} \\ {\text { (b) If } f^{-1}(3)=-1, \text { find } f(-1)}\end{array} $$

A linear function is given.(a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$ f(x)=\frac{1}{2} x+3 $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=x^{2}-1 $$

Sketch the graph of the function by first making a table of values. \(g(x)=-\sqrt{x}\)

Evaluate the function at the indicated values. $$ \begin{array}{l}{g(x)=\frac{1-x}{1+x}} \\ {g(2), g(-2), g\left(\frac{1}{2}\right), g(a), g(a-1), g(-1)}\end{array} $$

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{lll}{\text { (a) } f(f(4))} & {\text { (b) } g(g(3))}\end{array} $$

Assume that \(f\) is a one-to-one function. $$ \begin{array}{l}{\text { (a) If } f(5)=18, \text { find } f^{-1}(18)} \\\ {\text { (b) If } f^{-1}(4)=2, \text { find } f(2)}\end{array} $$

A linear function is given.(a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$ g(x)=-4 x+2 $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=x^{2}+5 $$

Sketch the graph of the function by first making a table of values. \(g(x)=\sqrt{-x}\)

Evaluate the function at the indicated values. $$ \begin{array}{l}{h(t)=t+\frac{1}{t}} \\ {h(1), h(-1), h(2), h\left(\frac{1}{2}\right), h(x), h\left(\frac{1}{x}\right)}\end{array} $$

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ g)(-2)} & {\text { (b) }(g \circ f)(-2)}\end{array} $$

If \(f(x)=5-2 x,\) find \(f^{-1}(3)\)

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=\sqrt{x}+1 $$

Sketch the graph of the function by first making a table of values. \(H(x)=|2 x|\)

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{2}-5 x $$

Evaluate the function at the indicated values. $$\begin{array}{l}{f(x)=2 x^{2}+3 x-4} \\ {f(0), f(2), f(-2), f(\sqrt{2}), f(x+1), f(-x)}\end{array}$$

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{lll}{\text { (a) }(f \circ f)(-1)} & {\text { (b) }(g \circ g)(2)}\end{array} $$

If \(g(x)=x^{2}+4 x\) with \(x \geq-2,\) find \(g^{-1}(5)\)

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=|x|-1 $$

Sketch the graph of the function by first making a table of values. \(H(x)=|x+1|\)

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{3}-4 x $$

Evaluate the function at the indicated values. $$\begin{array}{l}{f(x)=x^{3}-4 x^{2}} \\ {f(0), f(1), f(-1), f\left(\frac{3}{2}\right), f\left(\frac{x}{2}\right), f\left(x^{2}\right)}\end{array}$$

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ g)(x)} & {\text { (b) }(g \circ f)(x)}\end{array} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=x-6 ; \quad g(x)=x+6 $$

Population Growth and Decline The table gives the population in a small coastal community for the period \(1997-2006\) . Figures shown are for January 1 in each year. (a) What was the average rate of change of population between 1998 and 2001\(?\) (b) What was the average rate of change of population between 2002 and 2004\(?\) (c) For what period of time was the population increasing? (d) For what period of time was the population decreasing? $$ \begin{array}{|c|c|}\hline \text { Year } & {\text { Population }} \\ \hline 1997 & {624} \\ {1998} & {856} \\ {1999} & {1,336} \\ {2000} & {1,578} \\\ {2001} & {1,591} \\ {2002} & {1,483} \\ {2003} & {994} \\ {2004} & {826} \\\ {2005} & {801} \\ {2006} & {745} \\ \hline\end{array} $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=(x-5)^{2} $$

Sketch the graph of the function by first making a table of values. \(G(x)=|x|+x\)

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=2 x^{3}-3 x^{2}-12 x $$

Evaluate the function at the indicated values. $$ \begin{array}{l}{f(x)=2|x-1|} \\ {f(-2), f(0), f\left(\frac{1}{2}\right), f(2), f(x+1), f\left(x^{2}+2\right)}\end{array} $$

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ f)(x)} & {\text { (b) }(g \circ g)(x)}\end{array} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=3 x, \quad g(x)=\frac{x}{3} $$

Running Speed A man is running around a circular track that is \(200 \mathrm{~m}\) in circumference. An observer uses a stopwatch to record the runner's time at the end of each lap, obtaining the data in the following table. (a) What was the man's average speed (rate) between \(68 \mathrm{~s}\) and \(152 \mathrm{~s} ?\) (b) What was the man's average speed between \(263 \mathrm{~s}\) and \(412 \mathrm{~s} ?\) (c) Calculate the man's speed for each lap. Is he slowing down, speeding up, or neither? $$ \begin{array}{|c|c|} \hline \text { Time (s) } & \text { Distance (m) } \\ \hline 32 & 200 \\ 68 & 400 \\ 108 & 600 \\ 152 & 800 \\ 203 & 1000 \\ 263 & 1200 \\ 335 & 1400 \\ 412 & 1600 \\ \hline \end{array} $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=(x+1)^{2} $$

Sketch the graph of the function by first making a table of values. \(G(x)=|x|-x\)

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{4}-16 x^{2} $$

Evaluate the function at the indicated values. $$ \begin{array}{l}{f(x)=\frac{|x|}{x}} \\ {f(-2), f(-1), f(0), f(5), f\left(x^{2}\right), f\left(\frac{1}{x}\right)}\end{array} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=2 x-5 ; \quad g(x)=\frac{x+5}{2} $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=\sqrt{x+4} $$

Sketch the graph of the function by first making a table of values. \(f(x)=|2 x-2|\)

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{3}+2 x^{2}-x-2 $$

Evaluate the piecewise defined function at the indicated values. $$ \begin{array}{l}{f(x)=\left\\{\begin{array}{ll}{x^{2}} & {\text { if } x<0} \\\ {x+1} & {\text { if } x \geq 0}\end{array}\right.} \\ {f(-2), f(-1), f(0), f(1), f(2)}\end{array} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=\frac{3-x}{4} ; \quad g(x)=3-4 x $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=|x-3| $$

Sketch the graph of the function by first making a table of values. \(f(x)=\frac{X}{|x|}\)

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{4}-4 x^{3}+2 x^{2}+4 x-3 $$

Evaluate the piecewise defined function at the indicated values. $$ \begin{array}{ll}{f(x)=\left\\{\begin{array}{ll}{5} & {\text { if } x \leq 2} \\\ {2 x-3} & {\text { if } x>2}\end{array}\right.} \\ {f(-3), f(0), f(2),} & {f(3), f(5)}\end{array} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=\frac{1}{x} ; \quad g(x)=\frac{1}{x} $$

Cooling Soup When a bowl of hot soup is left in a room, the soup eventually cools down to room temperature. The temperature \(T\) of the soup is a function of time \(t\) . The table below gives the temperature (in "F) of a bowl of soup \(t\) minutes after it was set on the table. Find the average rate of change of the temperature of the soup over the first 20 minutes and over the next 20 minutes. During which interval did the soup cool off more quickly? $$ \begin{array}{|c|c|c|c|}\hline t(\min ) & {T(\mathbf{F})} & {t(\min )} & {T(\mathbf{F})} \\ \hline 0 & {200} & {35} & {94} \\ {5} & {172} & {40} & {89} \\ {10} & {150} & {50} & {81} \\ {15} & {133} & {60} & {77} \\ {20} & {119} & {90} & {72} \\ {25} & {108} & {120} & {70} \\ {30} & {100} & {150} & {70} \\ \hline\end{array} $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=-x^{3} $$