Chapter 1

In Exercises \(1-4,\) use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function. \begin{equation}f(x)=x^{4}-7 x^{2}+6 x\end{equation} \begin{equation} \begin{array}{ll}{\text { a. }[-1,1] \text { by }[-1,1]} & {\text { b. }[-2,2] \text { by }[-5,5]} \\ {\text { c. }[-10,10] \text { by }[-10,10]} & {\text { d. }[-5,5] \text { by }[-25,15]}\end{array} \end{equation}

\begin{array}{l}{\text { In Exercises } 1 \text { and } 2, \text { find the domains and ranges of } f, g, f+g, \text { and }} \\ {f \cdot g .}\end{array} $$f(x)=x, \quad g(x)=\sqrt{x-1}$$

On a circle of radius 10 \(\mathrm{m}\) , how long is an arc that subtends a central angle of (a) 4\(\pi / 5\) radians? (b) \(110^{\circ} ?\)

In Exercises \(1-6,\) find the domain and range of each function. $$f(x)=1+x^{2}$$

Use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function. \(\begin{equation} f(x)=x^{3}-4 x^{2}-4 x+16 \end{equation} \begin{equation} \begin{array}{ll}{\text { a. }[-1,1] \text { by }[-5,5]} & {\text { b. }[-3,3] \text { by }[-10,10]} \\ {\text { c. }[-5,5] \text { by }[-10,20]} & {\text { d. }[-20,20] \text { by }[-100,100]}\end{array} \end{equation}\)

In Exercises 1 and \(2,\) find the domains and ranges of \(f, g, f+g,\) and \(f \cdot g .\) $$f(x)=\sqrt{x+1}, \quad g(x)=\sqrt{x-1}$$

A central angle in a circle of radius 8 is subtended by an are of length 10\(\pi .\) Find the angle's radian and degree measures.

In Exercises \(1-6,\) find the domain and range of each function. $$f(x)=1-\sqrt{x}$$

Use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function. \begin{equation} f(x)=5+12 x-x^{3} \end{equation} \begin{equation} \begin{array}{ll}{\text { a. }[-1,1] \text { by }[-1,1]} & {\text { b. }[-5,5] \text { by }[-10,10]} \\ {\text { c. }[-4,4] \text { by }[-20,20]} & {\text { d. }[-4,5] \text { by }[-15,25]}\end{array} \end{equation}

In Exercises 3 and \(4,\) find the domains and ranges of \(f, g, f / g,\) and \(g / f .\) $$ f(x)=2, \quad g(x)=x^{2}+1 $$

You want to make an \(80^{\circ}\) angle by marking an arc on the perimeter of a 12 -in.-diameter disk and drawing lines from the ends of the arc to the disk's center. To the nearest tenth of an inch, how long should the arc be?

In Exercises \(1-6,\) find the domain and range of each function. $$ F(x)=\sqrt{5 x+10} $$

Use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function. \begin{equation} f(x)=\sqrt{5+4 x-x^{2}} \end{equation} \begin{equation} \begin{array}{ll}{\text { a. }[-2,2] \text { by }[-2,2]} & {\text { b. }[-2,6] \text { by }[-1,4]} \\ {\text { c. }[-3,7] \text { by }[0,10]} & {\text { d. }[-10,10] \text { by }[-10,10]}\end{array} \end{equation}

In Exercises 3 and \(4,\) find the domains and ranges of \(f, g, f / g,\) and \(g / f .\) $$ f(x)=1, \quad g(x)=1+\sqrt{x} $$

If you roll a 1 -m-diameter wheel forward 30 \(\mathrm{cm}\) over level ground, through what angle will the wheel turn? Answer in radians (to the nearest tenth) and degrees (to the nearest degree).

In Exercises \(1-6,\) find the domain and range of each function. $$g(x)=\sqrt{x^{2}-3 x}$$

If \(f(x)=x+5\) and \(g(x)=x^{2}-3,\) find the following. $$(a) f(g(0)) \quad \text { b. } g(f(0)) $$ $$ (c)f(g(x)) \quad \text { d. } g(f(x)) $$ $$ \begin{array}{ll}{\text { e. } f(f(-5))} & {\text { f. } g(g(2))} \\ {\text { g. } f(f(x))} & {\text { h. } g(g(x))}\end{array} $$

Copy and complete the following table of function values. If the function is undefined at a given angle, enter "UND." Do not use a calculator or tables. $$ \begin{array}{l}{\quad\boldsymbol{\theta} \quad-\pi \quad-2 \pi / 3 \quad 0 \quad \pi / 2 \quad 3 \pi / 4} \\ {\sin \theta} \\ {\cos \theta} \\ {\tan \theta} \\ {\cot \theta} \\ {\sec \theta} \\ {\csc \theta}\end{array} $$

In Exercises \(1-6,\) find the domain and range of each function. $$f(t)=\frac{4}{3-t}$$

If \(f(x)=x-1\) and \(g(x)=1 /(x+1),\) find the following. $$ \begin{array}{ll}{\text { a. } f(g(1 / 2))} & {\text { b. } g(f(1 / 2))} \\\ {\text { c. } f(g(x))} & {\text { d. } g(f(x))}\end{array} $$ $$ \begin{array}{ll}{\text { e. } f(f(2))} & {\text { f. } g(g(2))} \\ {\text { g. } f(f(x))} & {\text { h. } g(g(x))}\end{array} $$

Copy and complete the following table of function values. If the function is undefined at a given angle, enter "UND." Do not use a calculator or tables. $$ \begin{array}{l}{\quad\boldsymbol{\theta} \quad-3 \pi / 2 \quad-\pi / 3 \quad-\pi / 6 \quad \pi / 4 \quad 5 \pi / 6} \\ {\sin \theta} \\ {\cos \theta} \\ {\tan \theta} \\ {\cot \theta} \\ {\sec \theta} \\ {\csc \theta}\end{array} $$

In Exercises \(1-6,\) find the domain and range of each function. $$G(t)=\frac{2}{t^{2}-16}$$

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=x+1, \quad g(x)=3 x, \quad h(x)=4-x $$

In Exercises \(7-12,\) one of sin \(x, \cos x,\) and tan \(x\) is given. Find the other two if \(x\) lies in the specified interval. $$\sin x=\frac{3}{5}, \quad x \in\left[\frac{\pi}{2}, \pi\right]$$

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=3 x+4, \quad g(x)=2 x-1, \quad h(x)=x^{2} $$

In Exercises \(7-12,\) one of sin \(x, \cos x,\) and tan \(x\) is given. Find the other two if \(x\) lies in the specified interval. $$\tan x=2, \quad x \in\left[0, \frac{\pi}{2}\right]$$

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=\sqrt{x+1}, \quad g(x)=\frac{1}{x+4}, \quad h(x)=\frac{1}{x} $$

In Exercises \(7-12,\) one of sin \(x, \cos x,\) and tan \(x\) is given. Find the other two if \(x\) lies in the specified interval. $$\cos x=\frac{1}{3}, \quad x \in\left[-\frac{\pi}{2}, 0\right]$$

Express the area and perimeter of an equilateral triangle as a function of the triangle's side length \(x\) .

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. \(\begin{equation} f(x)=x^{2}\left(6-x^{3}\right) \end{equation}\)

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=\frac{x+2}{3-x}, \quad g(x)=\frac{x^{2}}{x^{2}+1}, \quad h(x)=\sqrt{2-x} $$

In Exercises \(7-12,\) one of sin \(x, \cos x,\) and tan \(x\) is given. Find the other two if \(x\) lies in the specified interval. $$\cos x=-\frac{5}{13}, \quad x \in\left[\frac{\pi}{2}, \pi\right]$$

Express the side length of a square as a function of the length \(d\) of the square's diagonal. Then express the area as a function of the diagonal length.

Let \(f(x)=x-3, \quad g(x)=\sqrt{x}, \quad h(x)=x^{3},\) and \(j(x)=2 x .\) Express each of the functions in Exercises 11 and 12 as a composite involving one or more of \(f, g, h,\) and \(j .\) $$\begin{array}{ll}{\text { a. } y=\sqrt{x}-3} & {\text { b. } y=2 \sqrt{x}} \\\ {\text { c. } y=x^{1 / 4}} & {\text { d. } y=4 x} \\ {\text { e. } y=\sqrt{(x-3)^{3}}} & {\text { f. } y=(2 x-6)^{3}}\end{array}$$

In Exercises \(7-12,\) one of sin \(x, \cos x,\) and tan \(x\) is given. Find the other two if \(x\) lies in the specified interval. $$\tan x=\frac{1}{2}, \quad x \in\left[\pi, \frac{3 \pi}{2}\right]$$

Express the edge length of a cube as a function of the cube's diagonal length \(d .\) Then express the surface area and volume of the cube as a function of the diagonal length.

Let \(f(x)=x-3, \quad g(x)=\sqrt{x}, \quad h(x)=x^{3},\) and \(j(x)=2 x .\) Express each of the functions in Exercises 11 and 12 as a composite involving one or more of \(f, g, h,\) and \(j .\) $$ \begin{array}{ll}{\text { a. } y=2 x-3} & {\text { b. } y=x^{3 / 2}} \\\ {\text { c. } y=x^{9}} & {\text { d. } y=x-6} \\ {\text { e. } y=2 \sqrt{x-3}} & {\text { f. } y=\sqrt{x^{3}-3}}\end{array} $$

In Exercises \(7-12,\) one of sin \(x, \cos x,\) and tan \(x\) is given. Find the other two if \(x\) lies in the specified interval. $$\sin x=-\frac{1}{2}, \quad x \in\left[\pi, \frac{3 \pi}{2}\right]$$

A point \(P\) in the first quadrant lies on the graph of the function \(f(x)=\sqrt{x} .\) Express the coordinates of \(P\) as functions of the slope of the line joining \(P\) to the origin.

Copy and complete the following table. $$\begin{array}{ll}{g(x)} & {f(x)} & {(f \circ g)(x)} \\ \hline\end{array}$$ $$ \begin{array}{ll}{\text { a. } x-7} & {\sqrt{x}} \\ {\text { b. } x+2} & {3 x} \\ {\text { c. } ?} & {\sqrt{x-5}} & {\sqrt{x^{2}-5}}\end{array} $$ $$ \text { d. } \frac{x}{x-1} \quad \frac{x}{x-1} \quad ? $$ $$ \text { e. }? \quad 1+\frac{1}{x} \quad x $$ $$ \begin{array}{ll}{\text { f. } \frac{1}{x}} & {?} & {x}\end{array} $$

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. \begin{equation} y=5 x^{2 / 5}-2 x \end{equation}

Consider the point \((x, y)\) lying on the graph of the line \(2 x+4 y=5 .\) Let \(L\) be the distance from the point \((x, y)\) to the origin \((0,0) .\) Write \(L\) as a function of \(x .\)

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. \begin{equation} y=x^{2 / 3}(5-x) \end{equation}

Consider the point \((x, y)\) lying on the graph of \(y=\sqrt{x-3}\) . Let \(L\) be the distance between the points \((x, y)\) and \((4,0) .\) Write \(L\) as a function of \(y .\)

Evaluate each expression using the functions $$ f(x)=2-x, \quad g(x)=\left\\{\begin{array}{lr}{-x,} & {-2 \leq x<0} \\ {x-1,} & {0 \leq x \leq 2}\end{array}\right. $$ $$ \begin{array}{lll}{\text { a. }} & {f(g(0))} & {\text { b. } g(f(3))} & {\text { c. } g(g(-1))} \\ {\text { d. }} & {f(f(2))} & {\text { e. } g(f(0))} & {\text { f. } f(g(1 / 2))}\end{array} $$

Find the natural domain and graph the functions in Exercises \(15-20 .\) $$f(x)=5-2 x$$

Find the natural domain and graph the functions in Exercises \(15-20 .\) $$f(x)=1-2 x-x^{2}$$

In Exercises 17 and \(18,\) (a) write formulas for \(f \circ g\) and \(g \circ f\) and find the \((\mathbf{b})\) domain and \((\mathbf{c})\) range of each. $$ f(x)=\sqrt{x+1}, g(x)=\frac{1}{x} $$

Find the natural domain and graph the functions in Exercises \(15-20 .\) $$g(x)=\sqrt{|x|}$$

In Exercises 17 and \(18,\) (a) write formulas for \(f \circ g\) and \(g \circ f\) and find the \((\mathbf{b})\) domain and \((\mathbf{c})\) range of each. $$ f(x)=x^{2}, g(x)=1-\sqrt{x} $$