Chapter 3

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\sqrt[3]{x-1}\)

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(g(t)=t \sqrt{t+3}\)

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). Given the function \(f(x)=x^{2}-3 x:\) (a) Evaluate \(f(5)\). (b) Solve \(f(x)=4\).

For the following exercises, use the values listed in to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\ \hline \end{array} $$ Solve \(f(x)=3\).

Cities \(A\) and \(B\) are on the same east-west line. Assume that city A is located at the origin. If the distance from city \(\mathrm{A}\) to city \(\mathrm{B}\) is at least 100 miles and \(x\) represents the distance from city B to city \(\mathrm{A}\), express this using absolute value notation.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\left|x^{2}+7\right|\)

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(k(t)=3 t^{\frac{2}{3}}-t\)

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{ll}x+1 & \text { if } x<-2 \\ -2 x-3 & \text { if } x \geq-2\end{array}\right.\)

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). Given the function \(f(x)=\sqrt{x+2}\) (a) Evaluate \(f(7)\). (b) Solve \(f(x)=4\).

For the following exercises, use the values listed in to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\ \hline \end{array} $$ Find \(f^{-1}(0)\)

The true proportion \(p\) of people who give a favorable rating to Congress is \(8 \%\) with a margin of error of \(1.5 \%\). Describe this statement using an absolute value equation.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\frac{1}{(x-2)^{3}}\)

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(m(x)=x^{4}+2 x^{3}-12 x^{2}-10 x+4\)

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{ll}2 x-1 & \text { if } x<1 \\ 1+x & \text { if } x \geq 1\end{array}\right.\)

For the following exercises, use the values listed in to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\ \hline \end{array} $$ Solve \(f^{-1}(x)=7\).

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable \(x\) for the score.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\left(\frac{1}{2 x-3}\right)^{2}\)

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(n(x)=x^{4}-8 x^{3}+18 x^{2}-6 x+2\)

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using \(x\) as the diameter of the bearing, write this statement using absolute value notation.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\sqrt{\frac{2 x-1}{3 x+4}}\)

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{cl}3 & \text { if } x<0 \\ \sqrt{x} & \text { if } x \geq 0\end{array}\right.\)

For the following exercises, find the inverse function. Then, graph the function and its inverse. \(f(x)=\frac{3}{x-2}\)

The tolerance for a ball bearing is 0.01 . If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is \(x\) inches, express the tolerance using absolute value notation.

Let \(f(x)=\frac{1}{x}\). Find a number \(c\) such that the average rate of change of the function \(f\) on the interval \((1, c)\) is \(-\frac{1}{4}\).

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{ll}x^{2} & \text { if } x<0 \\ 1-x & \text { if } x>0\end{array}\right.\)

For the following exercises, find the inverse function. Then, graph the function and its inverse. \(f(x)=x^{3}-1\)

Let \(f(x)=\frac{1}{x}\). Find the number \(b\) such that the average rate of change of \(f\) on the interval \((2, b)\) is \(-\frac{1}{10}\)

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{rll}x^{2} & \text { if } & x<0 \\ x+2 & \text { if } & x \geq 0\end{array}\right.\)

For the following exercises, find the inverse function. Then, graph the function and its inverse. Find the inverse function of \(f(x)=\frac{1}{x-1} .\) Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read \(22,125 .\) Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{cl}x+1 & \text { if } x<1 \\ x^{3} & \text { if } x \geq 1\end{array}\right.\)

To convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit, we use the formula \(f(x)=\frac{9}{5} x+32\). Find the inverse function, if it exists, and explain its meaning.

A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3: 40 p.m. At this time, he started pumping gas into the tank. At exactly \(3: 44,\) the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{ccc}|x| & \text { if } & x<2 \\ 1 & \text { if } & x \geq 2\end{array}\right.\)

The circumference \(C\) of a circle is a function of its radius given by \(C(r)=2 \pi r\). Express the radius of a circle as a function of its circumference. Call this function \(r(C)\). Find \(r(36 \pi)\) and interpret its meaning.

Near the surface of the moon, the distance that an object falls is a function of time. It is given by \(d(t)=2.6667 t^{2},\) where \(t\) is in seconds and \(d(t)\) is in feet. If an object is dropped from a certain height, find the average velocity of the object from \(t=1\) to \(t=2\)

For the following exercises, given each function \(f,\) evaluate \(f(-3), \quad f(-2), \quad f(-1),\) and \(f(0) .\) \(f(x)=\left\\{\begin{array}{ll}x+1 & \text { if } x<-2 \\ -2 x-3 & \text { if } x \geq-2\end{array}\right.\)

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, \(t,\) in hours given by \(d(t)=50 t\). Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function \(t(d)\). Find \(t(180)\) and interpret its meaning.

For the following exercises, determine whether the function is odd, even, or neither. \(f(x)=3 x^{4}\)

For the following exercises, given each function \(f,\) evaluate \(f(-3), \quad f(-2), \quad f(-1),\) and \(f(0) .\) \(f(x)=\left\\{\begin{array}{ll}1 & \text { if } x \leq-3 \\ 0 & \text { if } x>-3\end{array}\right.\)

For the following exercises, determine whether the function is odd, even, or neither. \(g(x)=\sqrt{x}\)

For the following exercises, given each function \(f,\) evaluate \(f(-3), \quad f(-2), \quad f(-1),\) and \(f(0) .\) \(f(x)=\left\\{\begin{array}{cl}-2 x^{2}+3 & \text { if } x \leq-1 \\ 5 x-7 & \text { if } x>-1\end{array}\right.\)

For the following exercises, determine whether the function is odd, even, or neither. \(h(x)=\frac{1}{x}+3 x\)

For the following exercises, given each function \(f,\) evaluate \(f(-1), \quad f(0), \quad f(2),\) and \(f(4) .\) \(f(x)=\left\\{\begin{array}{ll}7 x+3 & \text { if } x<0 \\ 7 x+6 & \text { if } x \geq 0\end{array}\right.\)

For the following exercises, determine whether the function is odd, even, or neither. \(f(x)=(x-2)^{2}\)

For the following exercises, given each function \(f,\) evaluate \(f(-1), \quad f(0), \quad f(2),\) and \(f(4) .\) \(f(x)=\left\\{\begin{array}{cc}x^{2}-2 & \text { if } x<2 \\ 4+|x-5| & \text { if } x \geq 2\end{array}\right.\)

For the following exercises, determine whether the function is odd, even, or neither. \(g(x)=2 x^{4}\)

For the following exercises, given each function \(f,\) evaluate \(f(-1), \quad f(0), \quad f(2),\) and \(f(4) .\) \(f(x)=\left\\{\begin{array}{ccc}5 x & \text { if } & x<0 \\ 3 & \text { if } & 0 \leq x \leq 3 \\ x^{2} & \text { if } & x>3\end{array}\right.\)

For the following exercises, determine whether the function is odd, even, or neither. \(h(x)=2 x-x^{3}\)

For the following exercises, write the domain for the piecewise function in interval notation. \(f(x)=\left\\{\begin{array}{c}x+1 \text { if } x<-2 \\ -2 x-3 \text { if } x \geq-2\end{array}\right.\)