Section 1
Real Numbers, Functions, and Graphs
Use a calculator to find a rational number $r$ such that $\left|r-\pi^{2}\right|<10^{-4}$.
Which of $(a)-(1)$ are true for $a=-3$ and $b=2 ?$ $$ (a)\quad a < b \quad (\mathbf{b})|a|<|b| \quad (c) a b>0 $$ $$(d) \quad 3 a<3 b \quad \text { (e) }-4 a<-4 b \quad \text { (f) } \frac{1}{a}<\frac{1}{b}$$
In Exercises 3- 8, express the interval in terms of an inequality involving absolute value.$$[-2,2]$$
In Exercises 3 - 8, express the interval in terms of an inequality involving absolute value.$$(-4,4)$$
In Exercises 3 - 8, express the interval in terms of an inequality involving absolute value.$$(0,4)$$
In Exercises 3 - 8, express the interval in terms of an inequality involving absolute value.$$[-4,0]$$
In Exercises 3 - 8, express the interval in terms of an inequality involving absolute value.$$[1,5]$$
9-12, write the inequality in the form $ a < x < b $.$$|x|<8$$
9-12, write the inequality in the form $ a < x < b $.$$|x-12|<8$$
9-12, write the inequality in the form $ a < x < b $.$$ |2 x+1|<5 $$
9-12, write the inequality in the form $ a < x < b $.$$ |3 x-4|<2 $$
In Exercises 13 - 18, express the set of numbers $x$ satisfying the given condition as an interval.$$|x|<4$$
In Exercises 13 - 18, express the set of numbers $x$ satisfying the given condition as an interval.$$|x| \leq 9$$
In Exercises 13 - 18, express the set of numbers $x$ satisfying the given condition as an interval.$$|x-4|<2$$
In Exercises 13 - 18, express the set of numbers $x$ satisfying the given condition as an interval.$$|x+7|<2$$
In Exercises 13 - 18, express the set of numbers $x$ satisfying the given condition as an interval.$$|4 x-1| \leq 8$$
In Exercises 13 - 18, express the set of numbers $x$ satisfying the given condition as an interval.$$|3 x+5|<1$$
In Exercises 19 - 22, describe the set as a union of finite or infinite intervals.$$|x :| x-4 |>2 \}$$
In Exercises 19 - 22, describe the set as a union of finite or infinite intervals.$$\{x :|2 x+4|>3\}$$
In Exercises 19 - 22, describe the set as a union of finite or infinite intervals.$$\left\{x :\left|x^{2}-1\right|>2\right\}$$
In Exercises 19 - 22, describe the set as a union of finite or infinite intervals.$$|x :| x^{2}+2 x |>2 \}$$
Match (a) $-(\mathrm{f})$ with $(\mathrm{i})-(\mathrm{vi})$.$$\begin{array}{ll}{\text { (a) } a>3} & {\text { (b) }|a-5|<\frac{1}{3}} \\ {\left(\text { c) }\left|a-\frac{1}{3}\right|<5\right.} & {\text { (d) }|a|>5} \\ {(\text { e) }|a-4|<3} & {\text { (f) } 1 \leq a \leq 5}\end{array}$$
Match (a) $-(\mathrm{f})$ with $(\mathrm{i})-(\mathrm{vi})$.$$\begin{array}{ll}{\text { (a) } a>3} & {\text { (b) }|a-5|<\frac{1}{3}} \\ {\left(\text { c) }\left|a-\frac{1}{3}\right|<5\right.} & {\text { (d) }|a|>5} \\ {(\text { e) }|a-4|<3} & {\text { (f) } 1 \leq a \leq 5}\end{array}$$ $$ \begin{array}{l}{\text { (i) } a \text { lies to the right of } 3 .} \\ {\text { (ii) } a \text { lies between } 1 \text { and } 7} \\ {\text { (iii) The distance from } a \text { to } 5 \text { is less than } \frac{1}{3}} \\ {\text { (iv) The distance from } a \text { to } 3 \text { is at most } 2 \text { . }}\end{array} $$ $$ \begin{array}{l}{\text { (v) } a \text { is less than } 5 \text { units from } \frac{1}{3}} \\ {\text { (vi) } a \text { lies either to the left of }-5 \text { or to the right of } 5 .}\end{array} $$
Describe $\left\{x : \frac{x}{x+1}<0\right\}$ as an interval. Hint: Consider the sign of $x$ and $x+1$ individually.
Describe the set of real numbers satisfying $|x-3|=|x-2|+1$ as a half-infinite interval.
Show that if $a>b,$ and $a, b \neq 0,$ then $b^{-1}>a^{-1},$ provided that $a$ and $b$ have the same sign. What happens if $a>0$ and $b<0 ?$
Which $x$ satisfies both $|x-3|<2$ and $|x-5|<1 ?$
Show that if $|a-5|<\frac{1}{2}$ and $|b-8|<\frac{1}{2},$ then $|(a+b)-13|<1 .$ Hint: Use the triangle inequality $(|a+b| \leq|a|+$ $|b| ) .$
Suppose that $$|x-4| \leq 1$$
Suppose that $$|x-4| \leq 1$$$$\begin{array}{l}{\text { (a) What is the maximum possible value of }|x+4| ?} \\ {\text { (b) Show that }\left|x^{2}-16\right| \leq 9}\end{array}$$
Suppose that $|a-6| \leq 2$ and $|b| \leq 3$.$$\begin{array}{l}{\text { (a) What is the largest possible value of }|a+b| ?} \\ {\text { (b) What is the smallest possible value of }|a+b| ?}\end{array}$$
Prove that $|x|-|y| \leq|x-y| .$ Hint: Apply the triangle inequality to $y$ and $x - y$.
Express $r_{1}=0 . \overline{27}$ as a fraction. Hint: $100 r_{1}-r_{1}$ is an integer.Then express $r_{2}=0.2666 \ldots$ as a fraction.
Represent 1$/ 7$ and 4$/ 27$ as repeating decimals.
The text states: If the decimal expansions of numbers a and b agree to $k$ places, then $|a-b| \leq 10^{-k}$ . Show that the converse is false: For all $k$ there are numbers $a$ and $b$ whose decimal expansions do not agree at all but $|a-b| \leq 10^{-k}$.
Plot each pair of points and compute the distance between them:$$\begin{array}{ll}{\text { (a) }(1,4) \text { and }(3,2)} & {\text { (b) }(2,1) \text { and }(2,4)} \\ {\text { (c) }(0,0) \text { and }(-2,3)} & {\text { (d) }(-3,-3) \text { and }(-2,3)}\end{array}$$
Find the equation of the circle with center $(2,4) :$$$\begin{array}{l}{\text { (a) with radius } r=3} \\ {\text { (b) that passes through }(1,-1)}\end{array}$$
Find all points in the $x y$ -plane with integer coordinates located at a distance 5 from the origin. Then find all points with integer coordinates located at a distance 5 from $(2,3) .$
Determine the domain and range of the function$$\begin{array}{c}{f :\{r, s, t, u\} \rightarrow\{A, B, C, D, E\}} \\ {\text { defined by } f(r)=A, f(s)=B, f(t)=B, f(u)=E}\end{array}$$
Give an example of a function whose domain $D$ has three elements and whose range $R$ has two elements. Does a function exist whose domain $D$ has two elements and whose range $R$ has three elements?
In Exercises 41 - 48, find the domain and range of the function.$$f(x)=-x$$
In Exercises 41 - 48, find the domain and range of the function.$$g(t)=t^{4}$$
In Exercises 41 - 48, find the domain and range of the function.$$f(x)=x^{3}$$
In Exercises 41 - 48, find the domain and range of the function.$$g(t)=\sqrt{2-t}$$
In Exercises 41 - 48, find the domain and range of the function.$$f(x)=|x|$$
In Exercises 41 - 48, find the domain and range of the function.$$h(s)=\frac{1}{s}$$
In Exercises 41 - 48, find the domain and range of the function.$$f(x)=\frac{1}{x^{2}}$$
In Exercises 41 - 48, find the domain and range of the function.$$g(t)=\cos \frac{1}{t}$$
In Exercises 49 - 52, determine where $f$ is increasing.$$f(x)=|x+1|$$
In Exercises 49 - 52, determine where $f$ is increasing.$$f(x)=x^{3}$$
In Exercises 49 - 52, determine where $f$ is increasing.$$f(x)=x^{4}$$
In Exercises 49 - 52, determine where $f$ is increasing.$$f(x)=\frac{1}{x^{4}+x^{2}+1}$$
In Exercises $53-58,$ find the zeros of $f$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.$$f(x)=x^{2}-4$$
In Exercises $53-58,$ find the zeros of $f$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.$$f(x)=2 x^{2}-4$$
In Exercises $53-58,$ find the zeros of $f$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.$$f(x)=x^{3}-4 x$$
In Exercises $53-58,$ find the zeros of $f$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.$$f(x)=x^{3}$$
In Exercises $53-58,$ find the zeros of $f$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.$$f(x)=2-x^{3}$$
In Exercises $53-58,$ find the zeros of $f$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.$$f(x)=\frac{1}{(x-1)^{2}+1}$$
Which of the curves in Figure 26 is the graph of a function?
Determine whether the function is even, odd, or neither.$${ (a) } f(x)=x^{5} \quad \text { (b) } g(t)=t^{3}-t^{2}$$$${(c)}F(t)=\frac{1}{t^{4}+t^{2}}$$
Determine whether the function is even, odd, or neither.$$\begin{array}{l}{\text { (a) } f(t)=\frac{1}{t^{4}+t+1}-\frac{1}{t^{4}-t+1} \text { (b) } g(t)=2^{t}-2^{-t}} \\ {\text { (c) } G(\theta)=\sin \theta+\cos \theta \quad \text { (d) } H(\theta)=\sin \left(\theta^{2}\right)}\end{array}$$
Write $f(x)=2 x^{4}-5 x^{3}+12 x^{2}-3 x+4$ as the sum of an even and an odd function.
Show that $f(x)=\ln \left(\frac{1-x}{1+x}\right)$ is an odd function.
State whether the function is increasing, decreasing, or neither.$$\begin{array}{l}{\text { (a) Surface area of a sphere as a function of its radius }} \\ {\text { (b) Temperature at a point on the equator as a function of time }} \\ {\text { (c) Price of an airline ticket as a function of the price of oil }} \\ {\text { (d) Pressure of the gas in a piston as a function of volume }}\end{array}$$
In Exercises $65-70,$ let $f$ be the function shown in Figure 27Find the domain and range of $f$
In Exercises $65-70,$ let $f$ be the function shown in Figure 27Sketch the graphs of $y=f(x+2)$ and $y=f(x)+2$
In Exercises $65-70,$ let $f$ be the function shown in Figure 27Sketch the graphs of $y=f(2 x), y=f\left(\frac{1}{2} x\right),$ and $y=2 f(x)$
In Exercises $65-70,$ let $f$ be the function shown in Figure 27Sketch the graphs of $y=f(-x)$ and $y=-f(-x)$
In Exercises $65-70,$ let $f$ be the function shown in Figure 27Extend the graph of $f$ to $[-4,4]$ so that it is an even function.
In Exercises $65-70,$ let $f$ be the function shown in Figure 27Extend the graph of $f$ to $[-4,4]$ so that it is an odd function.
Suppose that $f$ has domain $[4,8]$ and range $[2,6] .$ Find the domain and range of:$$\begin{array}{ll}{\text { (a) } y=f(x)+3} & {\text { (b) } y=f(x+3)} \\ {\text { (c) } y=f(3 x)} & {\text { (d) } y=3 f(x)}\end{array}$$
Let $f(x)=x^{2} .$ Sketch the graph over $[-2,2]$ of:$$\begin{array}{ll}{\text { (a) } y=f(x+1)} & {\text { (b) } y=f(x)+1} \\ {\text { (c) } y=f(5 x)} & {\text { (d) } y=5 f(x)}\end{array}$$
Suppose that the graph of $f(x)=\sin x$ is compressed horizontally by a factor of 2 and then shifted 5 units to the right.$$\begin{array}{l}{\text { (a) What is the equation for the new graph? }} \\ {\text { (b) What is the equation if you first shift by } 5 \text { and then compress by } 2 ?} \\ {\text { (c) } \mathrm{} \text { Verify your answers by plotting your equations. }}\end{array}$$
Figure 28 shows the graph of $f(x)=|x|+1 .$ Match the functions $(\mathrm{a})-(\mathrm{e})$ with their graphs $(\mathrm{i})-(\mathrm{v})$ .$$\begin{array}{ll}{\text { (a) } y=f(x-1)} & {\text { (b) } y=-f(x)} \\ {\text { (d) } y=f(x-1)-2} & {\text { (e) } y=f(x+1)}\end{array} \quad(\text { c) } y=-f(x)+2$$
Sketch the graph of $y=f(2 x)$ and $y=f\left(\frac{1}{2} x\right),$ where $f(x)=$ $|x|+1$ (Figure 28$) .$
Find the function $f$ whose graph is obtained by shifting the parabola $y=x^{2}$ by 3 units to the right and 4 units down, as in Figure $29 .$
Define $f(x)$ to be the larger of $x$ and $2-x .$ Sketch the graph of $f$ . What are its domain and range? Express $f(x)$ in terms of the absolute value function.
For each curve in Figure $30,$ state whether it is symmetric with respect to the $y$ -axis, the origin, both, or neither.
Show that the sum of two even functions is even and the sum of two odd functions is odd.
Suppose that $f$ and $g$ are both odd. Which of the following functions are even? Which are odd?$$\begin{array}{ll}{\text { (a) } y=f(x) g(x)} & {\text { (b) } y=f(x)^{3}} \\ {\text { (c) } y=f(x)-g(x)} & {\text { (d) } y=\frac{f(x)}{g(x)}}\end{array}$$
Prove that the only function whose graph is symmetric with respect to both the $y$ -axis and the origin is the function $f(x)=0 .$
Prove the triangle inequality $(|a+b| \leq|a|+|b|)$ by adding the two inequalities$$-|a| \leq a \leq|a|, \quad-|b| \leq b \leq|b|$$
Show that a fraction $r=a / b$ in lowest terms has a finite decimal expansion if and only if$$b=2^{n} 5^{m} \quad \text { for some } n, m \geq 0$$Hint: Observe that $r$ has a finite decimal expansion when $10^{N} r$ is an integer for some $N \geq 0$ (and hence $b$ divides $10^{N} ) .$
Let $p=p_{1} \ldots p_{s}$ be an integer with digits $p_{1}, \ldots, p_{s} .$ Show thatUse this to find the decimal expansion of $r=\frac{2}{11} .$ Note that$$r=\frac{2}{11}=\frac{18}{10^{2}-1}$$
A function $f$ is symmetric with respect to the vertical line $x=a$ if $f(a-x)=f(a+x)$$$\begin{array}{l}{\text { (a) Draw the graph of a function that is symmetric with respect to }} \\ {x=2 .} \\ {\text { (b) Show that if } f \text { is symmetric with respect to } x=a, \text { then } g(x)=} \\ {f(x+a) \text { is even. }}\end{array}$$
Formulate a condition for $f$ to be symmetric with respect to the point $(a, 0)$ on the $x$ -axis.