Chapter 1

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

In Exercises 1–4, use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. $$ \begin{array}{l}{f(x)=x^{4}-7 x^{2}+6 x} \\ {\text { a. }[-1,1] \text { by }[-1,1]} \\ {\text { c. }[-10,10] \text { by }[-10,10]}\end{array} \quad \text { b. }[-2,5] \text { by }[-5,5] $$

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

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

In Exercises \(1-4,\) identify each function as a constant function, linear function, power function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. Remember that some functions can fall into more than one category. $$ \begin{array}{ll}{\text { a. } f(x)=7-3 x} & {\text { b. } g(x)=\sqrt[5]{x}} \\\ {\text { c. } h(x)=\frac{x^{2}-1}{n^{2}+1}} & {\text { d. } r(x)=8^{x}}\end{array} $$

In Exercises \(1-4,\) a particle moves from \(A\) to \(B\) in the coordinate plane. Find the increments \(\Delta x\) and \(\Delta y\) in the particle's coordinates. Also find the distance from \(A\) to \(B\) . $$ A(-3,2), \quad B(-1,-2) $$

Express 1\(/ 9\) as a repeating decimal, using a bar to indicate the repeating digits. What are the decimal representations of 2\(/ 9 ? 3 / 9 ?\) 18\(/ 9 ? 9 / 9 ?\)

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

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

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

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

In Exercises \(1-4,\) identify each function as a constant function, linear function, power function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. Remember that some functions can fall into more than one category. $$ \begin{array}{ll}{\text { a. } F(t)=t^{4}-t} & {\text { b. } G(t)=5^{t}} \\\ {\text { c. } H(z)=\sqrt{z^{3}+1}} & {\text { d. } R(z)=\sqrt[3]{z^{7}}}\end{array} $$

In Exercises \(1-4,\) a particle moves from \(A\) to \(B\) in the coordinate plane. Find the increments \(\Delta x\) and \(\Delta y\) in the particle's coordinates. Also find the distance from \(A\) to \(B\) . $$ A(-1,-2), \quad B(-3,2) $$

Express 1\(/ 11\) as a repeating decimal, using a bar to indicate the repeating digits. What are the decimal representations of 2\(/ 11 ?\) 3\(/ 11 ? 9 / 11 ? 11 / 11 ?\)

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

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

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 \(1-4,\) identify each function as a constant function, linear function, power function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. Remember that some functions can fall into more than one category. $$ \begin{array}{ll}{\text { a. } y=\frac{3+2 x}{x-1}} & {\text { b. } y=x^{5 / 2}-2 x+1} \\ {\text { c. } y=\tan \pi x} & {\text { d. } y=\log _{7} x}\end{array} $$

In Exercises \(1-4,\) a particle moves from \(A\) to \(B\) in the coordinate plane. Find the increments \(\Delta x\) and \(\Delta y\) in the particle's coordinates. Also find the distance from \(A\) to \(B\) . $$ A(-3.2,-2), B(-8.1,-2) $$

If \(2 < x < 6,\) which of the following statements about \(x\) are necessarily true, and which are not necessarily true? $$ \begin{array}{ll}{\text { a. } 0 < x < 4} & {\text { b. } 0 < x-2 < 4} \\\ {\text { c. } 1 < \frac{x}{2} < 3} & {\text { d. } \frac{1}{6} < \frac{1}{x} < \frac{1}{2}} \\ {\text { e. } 1 < \frac{6}{x} < 3} & {\text { f. }|x-4| < 2} \\\ {\text { g. }-6 < -x < 2} & {\text { h. }-6 < -x < -2}\end{array} $$

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

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

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 \(1-4,\) identify each function as a constant function, linear function, power function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. Remember that some functions can fall into more than one category. $$ \begin{array}{ll}{\text { a. } y=\log _{5}\left(\frac{1}{t}\right)} & {\text { b. } f(z)=\frac{z^{5}}{\sqrt{z}+1}} \\ {\text { c. } g(x)=2^{1 / x}} & {\text { d. } w=5 \cos \left(\frac{t}{2}+\frac{\pi}{6}\right)}\end{array} $$

In Exercises \(1-4,\) a particle moves from \(A\) to \(B\) in the coordinate plane. Find the increments \(\Delta x\) and \(\Delta y\) in the particle's coordinates. Also find the distance from \(A\) to \(B\) . $$ A(\sqrt{2}, 4), \quad B(0,1.5) $$

If \(-1 < y-5 < 1,\) which of the following statements about \(y\) are necessarily true, and which are not necessarily true? $$ \begin{array}{ll}{\text { a. } 4 < y < 6} & {\text { b. }-6 < y<-4} \\ {\text { c. } y > 4} & {\text { d. } y < 6} \\ {\text { e. } 0 < y-4 < 2} & {\text { f. } 2 < \frac{y}{2}<3} \\ {\text { g. } \frac{1}{6} < \frac{1}{y} < \frac{1}{4}} & {\text { h. }|y-5| < 1}\end{array} $$

In Exercises \(1-6,\) find the domain and range of each function. $$ g(z)=\sqrt{4-z^{2}} $$

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

Describe the graphs of the equations in Exercises 5–8. $$ x^{2}+y^{2}=1 $$

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ -2 x>4 $$

In Exercises \(1-6,\) find the domain and range of each function. $$ g(z)=\frac{1}{\sqrt{4-z^{2}}} $$

In Exercises 5–30, determine an appropriate viewing window for the given function and use it to display its graph. $$ f(x)=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+1 $$

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

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ 8-3 x \geq 5 $$

If \(u(x)=4 x-5, v(x)=x^{2},\) and \(f(x)=1 / x,\) find formulas for the following. $$ \begin{array}{ll}{\text { a. } u(v(f(x)))} & {\text { b. } u(f(v(x)))} \\\ {\text { c. } v(u(f(x)))} & {\text { d. } v(f(u(x)))} \\ {\text { e. } f(u(v(x)))} & {\text { f. } f(v(u(x)))}\end{array} $$

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. $$ \sin x=\frac{3}{5}, \quad x \in\left[\frac{\pi}{2}, \pi\right] $$

Graph the functions in Exercises \(7-18 .\) What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$ y=-x^{3} $$

Describe the graphs of the equations in Exercises 5–8. $$ x^{2}+y^{2} \leq 3 $$

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ 5 x-3 \leq 7-3 x $$

If \(f(x)=\sqrt{x}, g(x)=x / 4,\) and \(h(x)=4 x-8,\) find formulas for the following. $$ \begin{array}{ll}{\text { a. } h(g(f(x)))} & {\text { b. } h(f(g(x)))} \\\ {\text { c. } g(h(f(x)))} & {\text { d. } g(f(h(x)))} \\ {\text { e. } f(g(h(x)))} & {\text { f. } f(h(g(x)))}\end{array} $$

In Exercises 5–30, determine an appropriate viewing window for the given function and use it to display its graph. $$ f(x)=4 x^{3}-x^{4} $$

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. $$ \tan x=2, \quad x \in\left[0, \frac{\pi}{2}\right] $$

Graph the functions in Exercises \(7-18 .\) What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$ y=-\frac{1}{x^{2}} $$

Describe the graphs of the equations in Exercises 5–8. $$ x^{2}+y^{2}=0 $$

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ 3(2-x)>2(3+x) $$

Consider the function \(y=\sqrt{(1 / x)-1}\) a. Can \(x\) be negative? b. \(\operatorname{Can} x=0 ?\) c. \(\operatorname{Can} x\) be greater than 1\(?\) d. What is the domain of the function?

Let \(f(x)=x-3, g(x)=\sqrt{x}, \quad h(x)=x^{3},\) and \(j(x)=2 x .\) Express each of the functions in Exercises 9 and 10 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. $$ \cos x=\frac{1}{3}, \quad x \in\left[-\frac{\pi}{2}, 0\right] $$