Section 1
1.1 Exercises
If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u}$, is it true that $f=g$ ?
If
$$f(x)=\frac{x^2-x}{x-1} \quad \text { and } \quad g(x)=x$$
is it true that $f=g$ ?
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?
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 $g$.
Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.
In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.
Determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.(GRAPH CAN'T COPY)
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?(GRAPH CAN'T COPY)
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.(GRAPH CAN'T COPY)
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.
Three runners compete in a 100 -meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?
The graph shows the power consumption for a day in September in San Francisco. ( $P$ is measured in megawatts; $t$ is measured in hours starting at midnight.)(a) What was the power consumption at 6 Am ? At 6 Pm ?(b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable?
Sketch a rough graph of the number of hours of daylight as a function of the time of year.
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
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.
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.
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.
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.
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 building, let $x(t)$ bethe 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.
The number $N$ (in millions) of US cellular phone subscribers is shown in the table. (Midyear estimates are given.)$$\begin{array}{|c|c|c|c|c|c|c|}\hline t & 1996 & 1998 & 2000 & 2002 & 2004 & 2006 \\\hline N & 44 & 69 & 109 & 141 & 182 & 233 \\\hline\end{array}$$(a) Use the data to sketch a rough graph of $N$ as a function of $t$.(b) Use your graph to estimate the number of cell-phone subscribers at midyear in 2001 and 2005.
Temperature readings $T$ (in ${ }^{\circ} \mathrm{F}$ ) were recorded every two hours from midnight to $2: 00 \mathrm{PM}$ in Phoenix on September 10, 2008. The time $t$ was measured in hours from midnight.$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\\hline T & 82 & 75 & 74 & 75 & 84 & 90 & 93 & 94 \\\hline\end{array}$$(a) Use the readings to sketch a rough graph of $T$ as a function of $t$.(b) Use your graph to estimate the temperature at 9:00 AM .
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 $f(a+h)$.
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.
Evaluate the difference quotient for the given function. Simplify your answer.$f(x)=4+3 x-x^2, \quad \frac{f(3+h)-f(3)}{h}$
Evaluate the difference quotient for the given function. Simplify your answer.$f(x)=x^3, \quad \frac{f(a+h)-f(a)}{h}$
Evaluate the difference quotient for the given function. Simplify your answer.$f(x)=\frac{1}{x}, \quad \frac{f(x)-f(a)}{x-a}$
Evaluate the difference quotient for the given function. Simplify your answer.$f(x)=\frac{x+3}{x+1}, \quad \frac{f(x)-f(1)}{x-1}$
Find the domain of the function.$f(x)=\frac{x+4}{x^2-9}$
Find the domain of the function.$f(x)=\frac{2 x^3-5}{x^2+x-6}$
Find the domain of the function.$f(t)=\sqrt[3]{2 t-1}$
Find the domain of the function.$g(t)=\sqrt{3-t}-\sqrt{2+t}$
Find the domain of the function.$h(x)=\frac{1}{\sqrt[4]{x^2-5 x}}$
Find the domain of the function.$f(u)=\frac{u+1}{1+\frac{1}{u+1}}$
Find the domain of the function.$F(p)=\sqrt{2-\sqrt{p}}$
Find the domain and range and sketch the graph of the function $h(x)=\sqrt{4-x^2}$.
Find the domain and sketch the graph of the function.$f(x)=2-0.4 x$
Find the domain and sketch the graph of the function.$F(x)=x^2-2 x+1$
Find the domain and sketch the graph of the function.$f(t)=2 t+t^2$
Find the domain and sketch the graph of the function.$H(t)=\frac{4-t^2}{2-t}$
Find the domain and sketch the graph of the function.$g(x)=\sqrt{x-5}$
Find the domain and sketch the graph of the function.$F(x)=|2 x+1|$
Find the domain and sketch the graph of the function.$G(x)=\frac{3 x+|x|}{x}$
Find the domain and sketch the graph of the function.$g(x)=|x|-x$
Find the domain and sketch the graph of the function.$f(x)= \begin{cases}x+2 & \text { if } x<0 \\ 1-x & \text { if } x \geqslant 0\end{cases}$
Find the domain and sketch the graph of the function.$f(x)= \begin{cases}3-\frac{1}{2} x & \text { if } x \leqslant 2 \\ 2 x-5 & \text { if } x>2\end{cases}$
Find the domain and sketch the graph of the function.$f(x)= \begin{cases}x+2 & \text { if } x \leqslant-1 \\ x^2 & \text { if } x>-1\end{cases}$
Find the domain and sketch the graph of the function.$f(x)= \begin{cases}x+9 & \text { if } x<-3 \\ -2 x & \text { if }|x| \leq 3 \\ -6 & \text { if } x>3\end{cases}$
Find an expression for the function whose graph is the given curve.The line segment joining the points $(1,-3)$ and $(5,7)$
Find an expression for the function whose graph is the given curve.The line segment joining the points $(-5,10)$ and $(7,-10)$
Find an expression for the function whose graph is the given curve.The bottom half of the parabola $x+(y-1)^2=0$
Find an expression for the function whose graph is the given curve.The top half of the circle $x^2+(y-2)^2=4$
Find an expression for the function whose graph is the given curve.(GRAPH CAN'T COPY)
Find a formula for the described function and state its domain.A rectangle has perimeter 20 m . Express the area of the rectangle as a function of the length of one of its sides.
Find a formula for the described function and state its domain.A rectangle has area $16 \mathrm{~m}^2$. Express the perimeter of the rectangle as a function of the length of one of its sides.
Find a formula for the described function and state its domain.Express the area of an equilateral triangle as a function of the length of a side.
Find a formula for the described function and state its domain.Express the surface area of a cube as a function of its volume.
Find a formula for the described function and state its domain.An open rectangular box with volume $2 \mathrm{~m}^3$ has a square base. Express the surface area of the box as a function of the length of a side of the base.
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft , express the area $A$ of the window as a function of the width $x$ of the window.
A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in . by 20 in . by cutting out equal squares of side $x$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $x$.
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 $0 \leqslant x \leqslant 600$.
In a certain state the maximum speed permitted on freeways is $65 \mathrm{mi} / \mathrm{h}$ and the minimum speed is $40 \mathrm{mi} / \mathrm{h}$. The fine for violating these limits is $$\$ 15$$ for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine $F$ as a function of the driving speed $x$ and graph $F(x)$ for $0 \leqslant x \leqslant 100$.
An electricity company charges its customers a base rate of $$\$ 10$$ a month, plus 6 cents per kilowatt-hour ( kWh ) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh . Express the monthly cost $E$ as a function of the amount $x$ of electricity used. Then graph the function $E$ for $0 \leqslant x \leqslant 2000$.
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 $I$.
The functions in Example 10 and Exercise 67 are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.
Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.(GRAPH CAN'T COPY)
(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?
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.
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$f(x)=\frac{x}{x^2+1}$
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$f(x)=\frac{x^2}{x^4+1}$
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$f(x)=\frac{x}{x+1}$
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$f(x)=x|x|$
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$f(x)=1+3 x^2-x^4$
Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$f(x)=1+3 x^3-x^5$
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.
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.