Problem 1
$$\begin{array}{l}{\text { If } f(x)=x+\sqrt{2-x} \text { and } g(u)=u+\sqrt{2-u}, \text { is it true }} \\ {\text { that } f=g^{\prime}}\end{array}$$
Problem 2
If $$f(x)=\frac{x^{2}-x}{x-1} \quad \text { and } \quad g(x)=x$$
is it true that f=g?
Problem 3
The graph of a function $f$ is given.
(a) State the value of $f(1) .$
(b) Estimate the value of $f(-1)$
(c) For what values of $x$ is $f(x)=1 ?$
(d) Estimate the value of $x$ such that $f(x)=0$ .
(e) State the domain and range of $f$ .
(f) On what interval is $f$ increasing?
Problem 4
The graphs of $f$ and $g$ are given.
(a) State the values of $f(-4)$ and $g(3) .$
(b) For what values of $x$ is $f(x)=g(x) ?$
(c) Estimate the solution of the equation $f(x)=-1$
(d) On what interval is $f$ decreasing?
(e) State the domain and range of $f .$
(f) State the domain and range of $g .$
Problem 5
Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 6
Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 7
Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 8
Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 9
The graph shown gives the weight of a certain person as a function of age. Describe in words how this person"s weight varies over time. What do you think happened when this person was 30 years old?
Problem 10
The graph shows the height of the water in a bathtub as a function of time. Give a verbal description of what you think happened.
Problem 11
You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.
Problem 12
Sketch a rough graph of the number of hours of daylight as a function of the time of year.
Problem 13
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
Problem 14
Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.
Problem 15
Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.
Problem 16
You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.
Problem 17
A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.
Problem 18
An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If $t$ represents the time in minutes since the plane has left the terminal, let $x(t)$ be the horizontal distance traveled and $y(t)$ be the altitude of the plane.
(a) Sketch a possible graph of $x(t) .$
(b) Sketch a possible graph of $y(t) .$
(c) Sketch a possible graph of the ground speed.
(d) Sketch a possible graph of the vertical velocity.
Problem 19
If $f(x)=3 x^{2}-x+2,$ find $f(2), f(-2), f(a), f(-a)$ $f(a+1), 2 f(a), f(2 a), f\left(a^{2}\right),[f(a)]^{2},$ and $f(a+h)$
Problem 20
A spherical balloon with radius $r$ inches has volume $V(r)=\frac{4}{3} \pi r^{3} .$ Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r+1$ inches.
Problem 21
Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=4+3 x-x^{2}, \quad \frac{f(3+h)-f(3)}{h}$$
Problem 22
Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=x^{3}, \quad \frac{f(a+h)-f(a)}{h}$$
Problem 23
Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{1}{x}, \quad \frac{f(x)-f(a)}{x-a}$$
Problem 24
Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{x+3}{x+1}, \quad \frac{f(x)-f(1)}{x-1}$$
Problem 30
Find the domain and range and sketch the graph of the function $h(x)=\sqrt{4-x^{2}}$
Problem 32
Find the domain and sketch the graph of the function.
$$F(x)=x^{2}-2 x+1$$
Problem 34
Find the domain and sketch the graph of the function.
$$H(t)=\frac{4-t^{2}}{2-t}$$
Problem 35
Find the domain and sketch the graph of the function.
$$g(x)=\sqrt{x-5}$$
Problem 37
Find the domain and sketch the graph of the function.
$$G(x)=\frac{3 x+|x|}{x}$$
Problem 39
Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {1-x} & {\text { if } x \geqq 0}\end{array}\right.$$
Problem 40
Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{3-\frac{1}{2} x} & {\text { if } x \leqslant 2} \\ {2 x-5} & {\text { if } x>2}\end{array}\right.$$
Problem 41
Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x \leqslant-1} \\ {x^{2}} & {\text { if } x>-1}\end{array}\right.$$
Problem 42
Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{-1} & {\text { if } x \leqslant-1} \\ {3 x+2} & {\text { if }|x|<1} \\ {7-2 x} & {\text { if } x \geqslant 1}\end{array}\right.$$
Problem 43
Find an expression for the function whose graph is the given curve.
The line segment joining the points $(1,-3)$ and $(5,7)$
Problem 44
Find an expression for the function whose graph is the given curve.
The line segment joining the points $(-5,10)$ and $(7,-10)$
Problem 45
Find an expression for the function whose graph is the given curve.
The bottom half of the parabola $x+(y-1)^{2}=0$
Problem 46
Find an expression for the function whose graph is the given curve.
The top half of the circle $x^{2}+(y-2)^{2}=4$
Problem 47
Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { A rectangle has perimeter } 20 \mathrm{m} \text { . Fxpress the area of the }} \\ {\text { rectangle as a function of the length of one of its sides. }}\end{array}$$
Problem 48
Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { A rectangle has area } 16 \mathrm{m}^{2} \text { . Express the perimeter of the }} \\ {\text { rectangle as a function of the length of one of its sides. }}\end{array}$$
Problem 49
Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { Express the area of an equilateral triangle as a function of }} \\ {\text { the length of a side. }}\end{array}$$
Problem 50
Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { Express the surface area of a cube as a function of its }} \\ {\text { volume. }}\end{array}$$
Problem 51
Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { An open rectangular box with volume } 2 \mathrm{m}^{3} \text { has a square }} \\ {\text { base. Express the surface area of the box as a function of }} \\ {\text { the length of a side of the base. }}\end{array}$$
Problem 52
A cell phone plan has a basic charge of $\$ 35$ a month. The
plan includes 400 free minutes and charges 10 cents for
each additional minute of usage. Write the monthly cost $C$
as a function of the number $x$ of minutes used and graph $C$
as a function of $x$ for 0$\leqslant x \leqslant 600 .$
Problem 53
In a certain country, income tax is assessed as follows.
There is no tax on income up to $\$ 10,000 .$ Any income over
$\$ 10,000$ is taxed at a rate of $10 \%,$ up to an income of
$\$ 20,000$ . Any income over $\$ 20,000$ is taxed at 15$\%$ .
(a) Sketch the graph of the tax rate $R$ as a function of the
income $I .$
(b) How much tax is assessed on an income of $\$ 14,000 ?$
On $\$ 26,000 ?$
(c) Sketch the graph of the total assessed tax $T$ as a function
of the income $I .$
Problem 54
The functions in Example 6 and Exercises 52 and 53$(\mathrm{a})$ are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.
Problem 55
Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.
Problem 56
Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.
Problem 57
(a) If the point $(5,3)$ is on the graph of an even function, what other point must also be on the graph?
(b) If the point $(5,3)$ is on the graph of an odd function, what other point must also be on the graph?
Problem 58
A function $f$ has domain $[-5,5]$ and a portion of its graph is shown.
(a) Complete the graph of $f$ if it is known that $f$ is even.
(b) Complete the graph of $f$ if it is known that $f$ is odd.
Problem 59
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x}{x^{2}+1}$$
Problem 60
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x^{2}}{x^{4}+1}$$
Problem 61
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x}{x+1}$$
Problem 62
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=x|x|$$
Problem 63
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=1+3 x^{2}-x^{4}$$
Problem 64
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=1+3 x^{3}-x^{5}$$
Problem 65
If $f$ and $g$ are both even functions, is $f+g$ even? If $f$ and $g$
are both odd functions, is $f+g$ odd? What if $f$ is even and
$g$ is odd? Justify your answers.
Problem 66
If $f$ and $g$ are both even functions, is the product $f g$ even $?$ If
$f$ and $g$ are both odd functions, is $f g$ odd? What if $f$ is even
and $g$ is odd? Justify your answers.