Chapter 3

A car moves along a straight test track. The distance traveled by the car at various times is shown in this table:$$\begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Time (seconds) } & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline \text { Distance (feet) } & 0 & 20 & 140 & 400 & 680 & 1400 & 1800 \\ \hline \end{array}$$ Find the average speed of the car over the interval from (a) 0 to 10 seconds (b) 10 to 20 seconds (c) 20 to 30 seconds (d) 15 to 30 seconds

Find \((f+g)(x),(f-g)(x),\) and \((g-f)(x)\) $$f(x)=-3 x+2, \quad g(x)=x^{3}$$

Find the indicated values of the function by hand and by using the table feature of a calculator (or the EVAL key on TI-85/86). If your answers do not agree with each other or with those at the back of the book, you are either making algebraic mistakes or incorrectly entering the function in the equation memory. \(f(x)=\frac{x-3}{x^{2}+4}\) (d) \(f(2)\) (a) \(f(-1)\) (b) \(f(0)\) (c) \(f(1)\) (e) \(f(3)\)

Determine whether or not the given table could possibly be a table of values of a function. Give reasons for your answer. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Input } & 1 & 0 & 3 & 1 & -5 \\ \hline \text { Output } & 2 & 3 & -2.5 & 2 & 14 \\ \hline \end{array}$$

Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=x^{4}-4 x+3$$

Find the average rate of change of the volume of the balloon in Example 2 as the radius increases from (a) 2 to 5 inches (b) 4 to 8 inches

Find \((f+g)(x),(f-g)(x),\) and \((g-f)(x)\) $$f(x)=x^{2}+2, \quad g(x)=x^{2}-4 x-2$$

Find the indicated values of the function by hand and by using the table feature of a calculator (or the EVAL key on TI-85/86). If your answers do not agree with each other or with those at the back of the book, you are either making algebraic mistakes or incorrectly entering the function in the equation memory. \(g(x)=\sqrt{x+4}-2\) (b) \(g(0)\) (c) \(g(4)\) (a) \(g(-2)\) (d) \(g(5)\) (e) \(g(12)\)

Determine whether or not the given table could possibly be a table of values of a function. Give reasons for your answer. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Input } & -5 & 3 & 0 & -3 & 5 \\ \hline \text { Output } & 0 & 3 & 0 & 5 & -3 \\ \hline \end{array}$$

Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=x^{3}+x-5$$

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$f(0)$$

Find \((f+g)(x),(f-g)(x),\) and \((g-f)(x)\) $$f(x)=1 / x, \quad g(x)=x^{2}+2 x-5$$

Determine whether or not the given table could possibly be a table of values of a function. Give reasons for your answer. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Input } & -5 & 1 & 3 & -5 & 7 \\ \hline \text { Output } & 0 & 2 & 4 & 6 & 8 \\ \hline \end{array}$$

Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=\left\\{\begin{array}{ll}x-3 & \text { if } x \leq 3 \\\2 x-6 & \text { if } x>3\end{array}\right.$$

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$f(1)$$

Find \((f+g)(x),(f-g)(x),\) and \((g-f)(x)\) $$f(x)=\sqrt{x}, \quad g(x)=x^{2}+1+\sqrt{x}$$

Determine whether or not the given table could possibly be a table of values of a function. Give reasons for your answer. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Input } & 1 & -1 & 2 & -2 & 3 \\ \hline \text { Output } & 1 & -2 & \pm 5 & -6 & 8 \\ \hline \end{array}$$

Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=x^{5}+2 x^{4}-x^{2}+4 x-5$$

The following table shows the total projected elementaryand secondary school enrollment (in thousands) for selected years. \(^{\dagger}\) Find the average rate of change of enrollmen from (a) 1980 to 1985 (b) 1985 to 1995 (c) 1995 to 2005 (d) 2005 to 2014 (e) During which of these periods was enrollment increasing at the fastest rate? At the slowest rate? $$\begin{array}{|c|c|} \hline \text { Year } & \text { Enrollment } \\ \hline 1980 & 40,877 \\ \hline 1985 & 39,422 \\ \hline 1990 & 41,217 \\ \hline 1995 & 44,840 \\ \hline 2000 & 47,204 \\ \hline 2005 & 48,375 \\ \hline 2010 & 48,842 \\ \hline 2014 & 49,993 \\ \hline \end{array}$$

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$f(\sqrt{2})$$

Deal with the greatest integer function of Example \(7,\) which is given by the equation \(y=[x]\). Compute the following values of the function: $$[6.75]$$

Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=x^{3}-4 x^{2}+x-10$$

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$f(\sqrt{2}-1)$$

Find \((f g)(x),(f / g)(x),\) and \((g / f)(x)\) $$f(x)=4 x^{2}+x^{4}, \quad g(x)=\sqrt{x^{2}+4}$$

Sketch the graph of the function, being sure to indicate which endpoints are included and which ones are excluded. $$f(x)=-[x]$$

Deal with the greatest integer function of Example \(7,\) which is given by the equation \(y=[x]\). Compute the following values of the function: $$[1.75]$$

Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=.1 x^{3}-.1 x^{2}-.005 x+1$$

The table shows the total number of shares traded (in billions) on the New York Stock Exchange in selected years. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & 1995 & 1997 & 1999 & 2001 & 2003 & 2005 \\ \hline \text { Volume } & 87.2 & 133.3 & 203.9 & 307.5 & 352.4 & 403.8 \\ \hline \end{array}$$Find the average rate of change in share volume from (a) 1995 to 1999 (b) 1999 to 2001 (c) 2001 to 2005 (d) 1995 to 2005 (e) During which of these periods did share volume crease at the fastest rate?

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$f(-2)$$

Find \((f g)(x),(f / g)(x),\) and \((g / f)(x)\) $$f(x)=x+5, \quad g(x)=x-5$$

Deal with the greatest integer function of Example \(7,\) which is given by the equation \(y=[x]\). Compute the following values of the function: $$[-4 / 3]$$

Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=.1 x^{3}+.005 x+1$$

Find \((f g)(x),(f / g)(x),\) and \((g / f)(x)\) $$f(x)=\sqrt{x^{2}-1}, \quad g(x)=\sqrt{x-1}$$

Sketch the graph of the function, being sure to indicate which endpoints are included and which ones are excluded. $$f(x)=\left\\{\begin{array}{ll}x^{2} & \text { if } x \geq-1 \\\2 x+3 & \text { if } x<-1\end{array}\right.$$

Deal with the greatest integer function of Example \(7,\) which is given by the equation \(y=[x]\). Compute the following values of the function: $$[5 / 3]$$

Use algebra to find the inverse of the given one-to-one function. $$f(x)=-x$$

Find the domains of fg and \(f / g .\) $$f(x)=x^{2}+1, \quad g(x)=1 / x$$

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$h(-4)$$

Use algebra to find the inverse of the given one-to-one function. $$f(x)=-x+1$$

When blood flows through an artery (which can be thought of as a cylindrical tube) its velocity is greatest at the center of the artery. Because of friction along the walls of the tube, the blood's velocity decreases as the distance \(r\) from the center of the artery increases, finally becoming 0 at the wall of the artery. The velocity (in centimeters per second) is given by the function \(v=18,500\left(.000065-r^{2}\right),\) where \(r\) is measured in centimeters. Find the average rate of change of the velocity as the distance from the center changes from (a) \(r=.001\) to \(r=.002\) (b) \(r=.002\) to \(r=.003\) (c) \(r=0\) to \(r=.005\)

Fill in the entries in the following table $$\begin{array}{|c|c|c|c|c|}\hline x & f(x) & g(x)=f(x)+2 & h(x)=\frac{1}{2} f(x) & i(x)=3 f(x)-2 \\\\\hline-1 & -1 / 2 & & & \\\\\hline 0 & 1 & & & \\\\\hline 1 & 2 & & & \\\\\hline 2 & 6 & & & \\\\\hline 3 & 8 & & & \\\\\hline\end{array}$$

Find the domains of fg and \(f / g .\) $$f(x)=x^{2}+2, \quad g(x)=\frac{1}{x^{2}+2}$$

Use algebra to find the inverse of the given one-to-one function. $$f(x)=5 x-4$$

Find the average rate of change of the function f over the given interval. $$f(x)=3+x^{3} \text { from } x=0 \text { to } x=2$$

Fill in the entries in the following table. If it is impossible to fill in an entry, put an X in it. $$\begin{array}{|c|c|c|c|c|}\hline t & f(t) & g(t)=f(t)-3 & h(t)=4 f(-t) & i(t)=f(t-1)-2 \\\\\hline-2 & 3 & & & \\\\\hline-1 & 6 & & & \\\\\hline 0 & 8 & & & \\\\\hline 1 & 0 & & & \\\\\hline 2 & 5 & & & \\\\\hline\end{array}$$

Find the domains of fg and \(f / g .\) $$f(x)=\sqrt{4-x^{2}}, \quad g(x)=\sqrt{3 x+4}$$

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$h(\pi+1)$$

Determine whether the equation defines \(y\) as a function of \(x\) or defines \(x\) as a function of \(y\) $$y=3 x^{2}-12$$

Use algebra to find the inverse of the given one-to-one function. $$f(x)=-3 x+5$$

Find the average rate of change of the function f over the given interval. $$f(x)=.25 x^{4}-x^{2}-2 x+4 \text { from } x=-1 \text { to } x=4$$