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Chapter 1

Functions and Models

Educators

JP
+ 8 more educators

Problem 1

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Heather Z.
Numerade Educator

Problem 2

If $ f(x) = \frac{x^2-x}{x-1} $ and $ g(x) = x $ is it true that $ f = g $?

Heather Z.
Numerade Educator

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?

Felicia S.
University of California - Los Angeles

Problem 4

The graph of a function $ 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 $.

JP
Jeffrey P.
Numerade Educator

Problem 5

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.

JP
Jeffrey P.
Numerade Educator

Problem 6

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.

JP
Jeffrey P.
Numerade Educator

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.

Heather Z.
Numerade Educator

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.

Heather Z.
Numerade Educator

Problem 9

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Heather Z.
Numerade Educator

Problem 10

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Heather Z.
Numerade Educator

Problem 11

Shown is a graph of the global average temperature $ T $ during the 20th century. Estimate the following.

(a) The global average temperature in 1950
(b) The year when the average temperature was 14.2 $ ^{\circ} C $
(c) The year when the temperature was smallest? Largest?
(d) The range of $ T $

JP
Jeffrey P.
Numerade Educator

Problem 12

Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000.

(a) What is the range of the ring width function?
(b) What does the graph tend to say about the temperature of the earth? Does the graph reflect the volcanic eruptions of the mid-19th century?

JP
Jeffrey P.
Numerade Educator

Problem 13

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.

JP
Jeffrey P.
Numerade Educator

Problem 14

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?

Heather Z.
Numerade Educator

Problem 15

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?

JP
Jeffrey P.
Numerade Educator

Problem 16

Sketch a rough graph of the number of hours of daylight as a function of the time of year.

Heather Z.
Numerade Educator

Problem 17

Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

Heather Z.
Numerade Educator

Problem 18

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.

Heather Z.
Numerade Educator

Problem 19

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.

Heather Z.
Numerade Educator

Problem 20

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.

Heather Z.
Numerade Educator

Problem 21

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.

Heather Z.
Numerade Educator

Problem 22

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) $ 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.

MC
Michael C.
Numerade Educator

Problem 23

Temperature readings $ T $ (in $ ^{\circ} $ F) were recorded every two hours from midnight to 2:00 PM in Atlanta on June 4, 2013. The time $ t $ was measured in hours from midnight.

(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

Heather Z.
Numerade Educator

Problem 24

Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks).

The table shows the data they obtained by averaging the BAC (in mg/mL) of the eight men.

(a) Use the readings to sketch the graph of the BAC as a function of $ t $.
(b) Use your graph to describe how the effect of alcohol varies with time.

JP
Jeffrey P.
Numerade Educator

Problem 25

If $ f(x) = 3x^2 - x + 2 $ , find $ f(2) $ , $ f(-2) $ , $ f(a) $ , $ f(-a) $ , $ f(a + 1) $ , 2 $ f(a) $ , $ f(2a) $ , $ f(a^2) $ , $ [ f(a) ]^2 $ , and $ f(a + h) $.

Heather Z.
Numerade Educator

Problem 26

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.

Heather Z.
Numerade Educator

Problem 27

Evaluate the difference quotient for the given function. Simplify your answer.

$ f(x) = 4 + 3x - x^2 $ , $ \dfrac{f(3 + h) - f(3)}{h} $

Heather Z.
Numerade Educator

Problem 28

Evaluate the difference quotient for the given function. Simplify your answer.

$ f(x) = x^3 $ , $ \dfrac{f(a + h) - f(a)}{h} $

Heather Z.
Numerade Educator

Problem 29

Evaluate the difference quotient for the given function. Simplify your answer.

$ f(x) = \dfrac{1}{x} $ , $ \dfrac{f(x) - f(a)}{x-a} $

Heather Z.
Numerade Educator

Problem 30

Evaluate the difference quotient for the given function. Simplify your answer.

$ f(x) = \dfrac{x + 3}{x + 1} $ , $ \dfrac{f(x) - f(1)}{x - 1} $

Heather Z.
Numerade Educator

Problem 31

Find the domain of the function.

$ f(x) = \dfrac{x + 4}{x^2 - 9} $

Heather Z.
Numerade Educator

Problem 32

Find the domain of the function.
$$ f(x) = \dfrac{2x^3 - 5}{x^2 + x - 6} $$

Heather Z.
Numerade Educator

Problem 33

Find the domain of the function.

$ f(t) = \sqrt[3]{2t - 1} $

Heather Z.
Numerade Educator

Problem 34

Find the domain of the function.

$ g(t) = \sqrt{3 - t} - \sqrt{2 + t} $

Heather Z.
Numerade Educator

Problem 35

Find the domain of the function.

$ h(x) = \dfrac{1}{\sqrt[4]{x^2 - 5x}} $

Heather Z.
Numerade Educator

Problem 36

Find the domain of the function.

$ f(u) = \dfrac{u + 1}{1 + \frac{1}{u + 1}} $

Heather Z.
Numerade Educator

Problem 37

Find the domain of the function.

$ F(p) = \sqrt{2 - \sqrt{p}} $

Heather Z.
Numerade Educator

Problem 38

Find the domain and range and sketch the graph of the function $ h(x) = \sqrt{4 - x^2} $.

Heather Z.
Numerade Educator

Problem 39

Find the domain and sketch the graph of the function.

$ f(x) = 1.6x - 2.4 $

Heather Z.
Numerade Educator

Problem 40

Find the domain and sketch the graph of the function.

$ g(t) = \dfrac{t^2 - 1}{t + 1} $

Heather Z.
Numerade Educator

Problem 41

Evaluate $ f(-3) $ , $ f(0) $ and $ f(2) $ for the piecewise defined function. Then sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
x + 2 & \mbox{if $ x < 0 $}\\
1 - x & \mbox{if $ x \ge 0 $}
\end{array} \right.$

Heather Z.
Numerade Educator

Problem 42

Evaluate $ f(-3) $ , $ f(0) $ and $ f(2) $ for the piecewise defined function. Then sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
3 - \frac{1}{2}x & \mbox{if $ x < 2 $}\\
2x - 5 & \mbox{if $ x \ge 2 $}
\end{array} \right.$

Heather Z.
Numerade Educator

Problem 43

Evaluate $ f(-3) $ , $ f(0) $ and $ f(2) $ for the piecewise defined function. Then sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
x + 1 & \mbox{if $ x \le -1 $}\\
x^2 & \mbox{if $ x > -1 $}
\end{array} \right.$

Heather Z.
Numerade Educator

Problem 44

Evaluate $ f(-3) $ , $ f(0) $ and $ f(2) $ for the piecewise defined function. Then sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
-1 & \mbox{if $ x \le 1 $}\\
7 - 2x & \mbox{if $ x >1 $}
\end{array} \right.$

Heather Z.
Numerade Educator

Problem 45

Sketch the graph of the function.

$ f(x) = x + | x | $

JP
Jeffrey P.
Numerade Educator

Problem 46

Sketch the graph of the function.

$ f(x) = | x + 2 | $

Heather Z.
Numerade Educator

Problem 47

Sketch the graph of the function.

$ g(t) = | 1 - 3t | $

Heather Z.
Numerade Educator

Problem 48

Sketch the graph of the function.

$ h(t) = | t | + | t + 1 | $

Felicia S.
University of California - Los Angeles

Problem 49

Sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
\mid x \mid & \mbox{ if $ | x | \le 1 $}\\
1 & \mbox{ if $ | x | > 1 $}
\end{array} \right.$

Heather Z.
Numerade Educator

Problem 50

Check back soon!

Problem 50

Sketch the graph of the function.

$ g(x) = || x | - 1 | $

Clarissa N.
Numerade Educator

Problem 51

Find an expression for the function whose graph is the given curve.

The line segment joining the points $ (1 , -3) $ and $ (5 , 7) $

Heather Z.
Numerade Educator

Problem 52

Find an expression for the function whose graph is the given curve.

The line segment joining the points $ (-5 , 10) $ and $ (7 , -10) $

Heather Z.
Numerade Educator

Problem 53

Find an expression for the function whose graph is the given curve.

The bottom half of the parabola $ x + (y - 1)^2 = 0 $

Heather Z.
Numerade Educator

Problem 54

Find an expression for the function whose graph is the given curve.

The top half of the circle $ x^2 + (y - 2)^2 = 4 $

Heather Z.
Numerade Educator

Problem 55

Find an expression for the function whose graph is the given curve.

Heather Z.
Numerade Educator

Problem 56

Find an expression for the function whose graph is the given curve.

Heather Z.
Numerade Educator

Problem 57

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.

Heather Z.
Numerade Educator

Problem 58

Find a formula for the described function and state its domain.

A rectangle has area $ 16 m^2 $. Express the perimeter of the rectangle as a function of the length of one of its sides.

Heather Z.
Numerade Educator

Problem 59

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.

Heather Z.
Numerade Educator

Problem 60

Find a formula for the described function and state its domain.

A closed rectangular box with volume $ 8 ft^3 $ has length twice the width. Express the height of the box as a function of the width.

Heather Z.
Numerade Educator

Problem 61

Find a formula for the described function and state its domain.

An open rectangular box with volume 2 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.

Heather Z.
Numerade Educator

Problem 62

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.

Heather Z.
Numerade Educator

Problem 63

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.

Heather Z.
Numerade Educator

Problem 64

A cell phone plan has a basic charge of 35 dollars 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 \le x \le 600 $.

Heather Z.
Numerade Educator

Problem 65

In a certain state the maximum speed permitted on freeways is 65 mi/h and the minimum speed is 40 mi/h. The fine for violating these limits is 15 dollars for every mile per hour above the maximum speed or below the maximum speed. Express the amount of the fine $ F $ as a function of the driving speed $ x $ and graph $ F(x) $ for $ 0 \le x \le 100 $.

JP
Jeffrey P.
Numerade Educator

Problem 66

An electricity company charges its customers a base rate of 10 dollars 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 \le x \le 2000 $.

JP
Jeffrey P.
Numerade Educator

Problem 67

In a certain country, income tax is assessed as follows. There is no tax on income up to 10,000 dollars. Any income over 10,000 dollars is taxed at a rate of 10%, up to an income of 20,000 dollars. Any income over 20,000 dollars 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 dollars? On 26,000 dollars?
(c) Sketch the graph of the total assessed tax $ T $ as a function of the income $ I $.

Clarissa N.
Numerade Educator

Problem 68

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.

JP
Jeffrey P.
Numerade Educator

Problem 69

Graphs of $ f $ and $ t $ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

Heather Z.
Numerade Educator

Problem 70

Graphs of $ f $ and $ t $ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

Heather Z.
Numerade Educator

Problem 71

(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?

Heather Z.
Numerade Educator

Problem 72

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.

Heather Z.
Numerade Educator

Problem 73

Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = \dfrac{x}{x^2 + 1} $

Heather Z.
Numerade Educator

Problem 74

Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = \dfrac{x^2}{x^4 + 1} $

Heather Z.
Numerade Educator

Problem 75

Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = \dfrac{x}{x + 1} $

Heather Z.
Numerade Educator

Problem 76

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 | $

Heather Z.
Numerade Educator

Problem 77

Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = 1 + 3x^2 - x^4 $

Heather Z.
Numerade Educator

Problem 78

Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = 1 + 3x^3 - x^5 $

Heather Z.
Numerade Educator

Problem 79

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.

Heather Z.
Numerade Educator

Problem 80

If $ f $ and $ g $ are both even functions, is the product $ fg $ even? If $ f $ and $ g $ are both odd functions, is $ fg $ odd? What if $ f $ is even and $ g $ is odd? Justify your answers.

Heather Z.
Numerade Educator