Stewart J. College Algebra: Concepts and Contexts

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6.1 Working with Functions: Algebraic Operations

■

Adding and Subtracting Functions

■

Multiplying and Dividing Functions

IN THIS SECTION… we study how functions can be combined using algebraic

operations: addition, subtraction, multiplication, and division.

It would be fun to throw a ball up into the air and have it keep going up indefinitely.

That’s exactly what would happen if there were no gravity. In that case, if you threw

a ball up at a speed of 48 feet per second, it would continue traveling at this same

speed indefinitely, so it would reach a height of feet in t seconds. But be-

cause of gravity, the ball is pulled down a distance feet in t seconds. To

find out how high the ball would actually reach in the presence of gravity, we need

to subtract the distance gravity pulls down on the ball from the distance the ball trav-

els up under the force of the initial throw. So gives the height the ball

reaches at time t. The situation is as follows.

■

Throwing a ball pushes it up a distance feet in t seconds, and at the

same time gravity pulls it down a distance feet.

■

The height the ball actually reaches in t seconds is given by .

So the actual height the ball reaches at time t is modeled by a combination of the two

functions f and (see Example 3). In this section we will see several other real-world sit-

uations in which a model is found by combining functions using arithmetic operations.

f 1t2- g1t 2

g1t 2

f 1t2

f 1t2- g1t 2

g1t 2= 16t

f 1t2= 48t

484 CHAPTER 6

■

Power, Polynomial, and Rational Functions

■ Adding and Subtracting Functions

Two functions f and can be combined to form new functions and in a

manner similar to the way in which we add and subtract numbers. For example, we

define the function by

The new function is called the sum of the functions f and ; its value at x is

. Of course, the sum on the right-hand side makes sense only if both

and are defined, that is, if x belongs to the domain of f and also to the domain of

. So if the domain of f is A and the domain of is B, then the domain of is the

intersection of these domains, that is, . We define the difference of the

functions f and in a similar way, as described in the following box.

Sums and Differences of Functions

f - gA 傽 B

f + ggg

g1x 2

f 1x2f 1x2+ g1x 2

gf + g

1f + g21x 2= f 1x 2+ g1x 2

f + g

f - gf + gg

Let f and be functions with domains A and B respectively. We define new

functions as follows:

■

The sum of f and is the function defined by

■

The difference of f and is the function defined by

Each of these functions has the domain .A 傽 B

1f - g21x 2= f 1x 2- g1x 2

f - gg

1f + g21x 2= f 1x 2+ g1x 2

f + gg

Image copyright originalpunkt, 2009. Used under license from Shutterstock.com

SECTION 6.1

■

Working with Functions: Algebraic Operations 485

example

Adding and Subtracting Functions

Let and .

(a) Find the functions and and their domains.

(b) Find and .

Solution

(a) The functions f and have domain ⺢, so the domain of is also ⺢.

Definition of

Definitions of f and

Simplify

The domain of is also ⺢. We have

Definition of

Definitions of f and

Distributive Property, simplify

(b) Each of these values exists because is in the domain of both functions.

From part (a) we have

1f - g2112= 1

- 2

1 - 4 =-5

1f + g2112= 1

+ 2

1 + 12 = 15

x = 1

= x

- 2x - 4

g = 1x

+ 42- 12x + 82

f - g 1f - g21x2= f 1x2- g1x 2

f - g

= x

+ 2x + 12

g = 1x

+ 42+ 12x + 82

f + g 1f + g21x2= f 1x2+ g1x 2

f + gg

f - gf + gg

1f - g211 21f + g211 2

f - gf + g

g1x 2= 2x + 8f 1x 2= x

+ 4

(a) Graph of f+g

f+g

(b) Graph of f-g

f-g

figure 1

■ NOW TRY EXERCISES 13 AND 15 ■

In general, the graph of the function can be obtained from the graphs of f

and by graphical addition. This means that we add corresponding y-coordinates,

as illustrated in the next example.

f + g

486 CHAPTER 6

■

Power, Polynomial, and Rational Functions

■ NOW TRY EXERCISE 21 ■

y=˝

y=Ï

figure 2

example

Graphical Subtraction

A ball is thrown straight up with a velocity of 48 feet per second. In the absence of

gravity the ball reaches a height in t seconds. Gravity pulls the ball down

feet in t seconds.

(a) Find a function h that models the height of the ball t seconds after it is thrown.

(b) Sketch a graph of the functions f, , and h. On what interval is the model h

valid?

Solution

(a) The function we want is

Ball moves up and down

Definitions of f and

(b) The functions f, , and h are graphed in Figure 4. The ball is at ground level

when . From the graph we see that at times 0 and 3 seconds.

The model is valid on the interval ; for numbers outside this interval we

see from the graph that the ball would be below ground level.

■ NOW TRY EXERCISE 23 ■

30, 34

h1t 2= 0h1t 2= 0

g = 48t - 16t

g1t 2f 1t 2 h1t 2= f 1t 2- g1t 2

g1t 2= 16t

f 1t2= 48t

f(x)

g(x)

y=(f+g)(x)

y=˝

y=Ï

f(x)

figure 3 Graphical addition of f and g

120

160

200

figure 4 Graphs of f, , and hg

example

Graphical Addition

The graphs of f and are shown in Figure 2. Use graphical addition to graph the func-

tion .

Solution

We obtain the graph of by “graphically adding” the values of and

as shown in Figure 3. This is implemented by copying the line segment PQ on top

of PR to obtain the point S on the graph of .f + g

g1x 2f 1x2f + g

f + g

SECTION 6.1

■

Working with Functions: Algebraic Operations 487

Combining functions occurs in many business models. For example, if the rev-

enue a company receives as a result of selling x units of their product is given by

and the cost of producing these x units is , then the profit is the difference

of these two functions.

The next example illustrates the process.

P1x 2= R1x2- C1x 2

P1x 2C1x 2

R1x 2

example

Finding Profit

A doughnut shop sells bear claws for 80 cents each, so the revenue from selling x

bear claws is dollars. The cost of producing x bear claws is modeled

by . These models are valid for x in the interval

(a) Find a function P that models the profit when x bear claws are produced and

sold.

(b) Sketch graphs of the functions R, C, and P. How many bear claws must be

sold before revenue exceeds cost?

Solution

(a) The profit function is the difference between the revenue and cost functions.

Definition of profit

Definitions of R and C

Distributive Property

Simplify

(b) Since the models are valid in the interval , we sketch graphs of R, C,

and P on that interval (see Figure 5).

30, 10004

=-50 + 0.40x + 0.0002x

= 0.80x - 50 - 0.40x + 0.0002x

= 0.80x - 150 + 0.40x - 0.0002x

P1x 2= R1x 2- C1x2

30, 10004

C1x 2= 50 + 0.40x - 0.0002x

R1x 2= 0.80x

The cost function C includes $50 of

fixed costs and $0.40 per unit

produced. The term

reflects the reduction in cost per

item when large quantities are

produced.

- 0.0002x

200

400 600 800 1000

100

200

300

400

500

figure 5 Graphs of R, C, and P

We see from the graph that revenue exceeds cost when x is greater than about

120. So about 120 bear claws must be produced and sold before a profit is

made. (Note from the graph that the values of P are positive when x is greater

than 120.)

■ NOW TRY EXERCISE 35 ■

488 CHAPTER 6

■

Power, Polynomial, and Rational Functions

example

Products and Quotients of Functions

Let and .

(a) Find the functions and and their domains.

(b) Find and .

Solution

(a) The domain of f is all real numbers, and the domain of is , so their

intersection is . We have

Definition of

Definitions of f and

The domain of is . Also,

Definition of

Definitions of f and

The domain of is . Note that in the domain of we exclude 0

because is 0.

(b) We can calculate each of these values because 4 is in the domain of each

function. Using the results of part (a), we have

■ NOW TRY EXERCISE 31 ■

b142=

4 + 1

1fg2142= 14 + 1214 = 10

g10 2

f>g5x 0 x 7 06f>g

g =

x + 1

fg a

b1x2=

f 1x2

g1x 2

5x 0 x Ú 06fg

g = 1x + 121x

fg 1fg21x2= f 1x 2

g1x 2

5x 0 x Ú 06

5x 0 x Ú 06g

1f>g21421f g214 2

f>gfg

g1x 2= 1xf 1x 2= x + 1

Let f and be functions with domains A and B. We define new functions as

follows:

■

The product of f and is the function defined by

The domain of is .

■

The quotient of f and is the function defined by

The domain of is ; that is, we must remove the

values of x where the denominator is 0.g1x2

5x H A 傽 B

0 g1x2⫽ 06f>g

b1x2=

f 1x2

g1x 2

f>gg

A 傽 Bfg

1f g21x2= f 1x2

g1x2

fgg

■ Multiplying and Dividing Functions

The functions f and can be combined to form new functions and in a man-

ner similar to the way in which we defined the sum and difference of two functions.

Products and Quotients of Functions

f>gf

SECTION 6.1

■

Working with Functions: Algebraic Operations 489

example

Products of Functions

An employee invests part of his monthly paycheck in his company’s stock. The table

gives the price (in dollars) of the stock on the purchase day and the number

of shares the employee buys.

(a) What is the meaning of the product function ?

(b) Calculate the values of for the indicated months. What do you conclude?R1t 2

R1t 2= N1t2P1t2

N1t 2P1t 2

Month t 1 2 3 4 5 6 7 8 9 10 11 12

Price P(t) 18 18 20 21 23 23 23 22 20 18 15 15

Number N(t) 10 10 12 12 12 12 12 15 20 25 30 30

Month t 1 2 3 4 5 6 7 8 9 10 11 12

R(t) 180 180 240 252 276 276 276 330 400 450 450 450

It appears that this employee is investing more money every month in his com-

pany’s stock.

■ NOW TRY EXERCISE 39 ■

6.1 Exercises

CONCEPTS

Fundamentals

1. Let f and be functions defined for all real numbers.

(a) By definition,

_______ and _______.

So if and , then

_______ and

_______.

(b) By definition,

_______ and _______. So if

and , then

_______ and _______.

2. Use the graphs of f and in the figure in the margin to find the following.

_______ _______

Think About It

3–4 ■ True or false?

3. If for all x, then the graph of is above the graph of for all x.

4. If for all x, then the graph of is below the graph of for all x.gfgf 1x 26 0

gf + gf 1x 27 0

1f>g2122=1f g212 2=

1f - g2122=1f + g212 2=

1f>g211 2=1f g211 2=g112= 6

f 11 2= 31f>g21x 2=1f

g21x2=

1f - g213 2=

1f + g213 2=g13 2= 7f 132= 5

1f - g21x 2=1f + g21x 2=

Solution

(a) The function R(t) gives the amount the employee invests in his company’s

stock each month.

(b) The value of R(t) at any t is the product of P(t) and N(t). Therefore

. The other entries in the table below are

calculated similarly.

10 = 180R11 2= N112P112=

t 0 1 2 3 4

f 1t 2

30 24 22 17 13

g1t 2

294 312 331 352 370

1f ⴙ g 21t2

1f ⴚ g 21t2

490 CHAPTER 6

■

Power, Polynomial, and Rational Functions

5–6

■ Functions f and are given in the table. Complete the table to find and .

f - gf + gg

SKILLS

t 0 1 2 3 4

f 1t 2

131 282 291 332 504

g1t 2

459 376 367 274 209

1f ⴙ g 21t2

1f ⴚ g 21t2

7–12

■ Use the graphs of f and shown in the margin to evaluate the expression.

7. 8.

9. 10.

11. 12.

13–14

■ The functions f and are given.

(a) Find the function . Draw graphs of f, , and in the same viewing

rectangle.

(b) Find the function . Draw graphs of f, , and in the same viewing

rectangle.

13.

14.

15–20

■ The functions f and are given.

(a) Find the functions and and their domains.

(b) Evaluate the functions and at the indicated value of x, if defined.

15. ,;116. ,;

17. ,;218. ,;1

19. ,;320. ,;5

21–22

■ Use graphical addition to sketch the graph of .

21. 22.

f + g

g1x 2= 21 + x

f 1x 2= 24 - xg1x 2= 23 + xf 1x2= 27 + x

g1x 2= 3x

- 1f 1x 2= x

+ 2xg1x2= 2 - 7x

f 1x 2= x

- 5x

- 1g1x 2= x + 2f 1x2= 0.5x

g1x 2= x - 3f 1x2= x

f - gf + g

f 1x 2= 3x

+ 5,g1x2= 4x + 7

f 1x 2= 2x

+ 1,g1x2= 3x + 2

f - ggf - g

f + ggf + g

1f g21421f>g2102

1g>f 210 21fg2112

1f - g210 21f + g212 2

t 0 1 2 3 4

f 1t 2

- 3 - 0.5

3 5 6

g 1t 2

2 6 7

- 5 - 2

1f g 21t2

1f>g 21t 2

t 0 1 2 3 4

f 1t 2

- 1

9 0 5

- 2

g 1t 2

- 21 - 25 - 11

0 1

1f g 21t2

1f>g 21t 2

SECTION 6.1

■

Working with Functions: Algebraic Operations 491

23–26 ■ The functions f and are given.

(a) Draw the graphs of f, , and on a common screen to illustrate graphical

addition.

(b) Draw the graphs of f, , and on a common screen to illustrate graphical

subtraction.

23. , 24. ,

25. , 26. ,

27–28

■ Functions f and are given in the table. Use the table to find and , if defined.

27.

f>gfgg

g1x 2= 0.5xf 1x2= x

g1x 2= 11 - 0.5xf 1x 2= 11 + 0.5x

g1x 2= x

f 1x 2= x

g1x 2= xf 1x 2= x

f - gg

f + gg

28.

y=C(x)

y=R(x)

100

2000

CONTEXTS

29–34 ■ The functions f and are given.

(a) Find the functions and and their domains.

(b) Evaluate the functions and at the indicated value, if defined.

29. ,;230. ,;

31. ,;032. ,;

33. ,;434. ,;0

35. Revenue, Cost, and Profit A store sells a certain type of digital camera. The revenue

from selling x cameras is modeled by . The cost of producing x

cameras is modeled by . These models are valid for x in

the interval .

(a) Find a function that models the profit from producing and selling x cameras.

(b) Sketch a graph of the revenue, cost and profit functions. How many cameras must

be sold in one week before revenue exceeds cost.

36. Revenue, Cost, and Profit The figure in the margin shows graphs of the cost and

revenue functions reported by a manufacturer of tires.

(a) Identify on the graph the value of x for which the profit is 0.

(b) Use graphical addition to sketch a graph of the profit function.

30, 4504

C1x 2= 9300 + 150x - 0.1x

R1x 2= 450x - 0.5x

g1x 2=

x + 1

f 1x 2=

x + 1

g1x 2=

x + 4

f 1x 2=

- 2g1x 2= 4x + 8f 1x 2= 3x

g1x 2= x

f 1x 2= x - 3

- 1g1x2= 5x + 10f 1x2= 2x - 1g1x 2= x + 5f 1x 2= 3x + 4

f>gfg

492 CHAPTER 6

■

Power, Polynomial, and Rational Functions

37–38

■ Revenue, Cost, and Profit A print shop makes bumper stickers for election

campaigns. If x stickers are ordered (where ), then the price per sticker

is dollars, and the total cost of producing the order is

dollars.

37. Use the fact that to express R(x), the

revenue from an order of x stickers, as a product of two functions of x.

38. Use the fact that to express P(x), the profit on an order of

x stickers, as a difference of two functions of x.

39. Home Sales Let be the number of existing homes sold in the United States in

month t of 2007, and let be the same function as f but for 2008. The table gives the

values of f and .

(a) What is the meaning of the function ?

(b) What is the meaning of the function ?

R1t 2=

h1t 2

f 1t 2

h1t 2= 1f - g21t2

f 1t 2

profit = revenue - cost

revenue = price per item * number of items sold

0.095x - 0.0000005x

0.15 - 0.000002x

x 6 10,000

41. Smoking in the United States The table below shows estimated numbers of

smokers as well as the total population of the United States from 1970 to 2007.

(a) What is the meaning of the quotient function ?

(b) Calculate the values of for the indicated years. What do you conclude?R1t 2

R1t 2=

S1t 2

P1t 2

Month t 1 2 3 4 5 6 7 8 9 10 11 12

2007 sales: f(t)

(thousands)

532 550 509 494 494 479 480 458 426 422 418 409

2008 sales: g(t)

(thousands)

408 419 412 408 416 404 418 409 428 409 370 396

40. Labor Force The following table shows the number of workers in the U.S. labor

force and the number of employed workers for each year from 2001 to 2008, as

reported by the Bureau of Labor Statistics from the Current Population Survey.

(a) What is the meaning of the function ?

(b) What is the meaning of the quotient functions and ?

conclude?

1t 2R

1t 2

1t 2=

U1t 2

L1t 2

1t 2=

E1t 2

L1t 2

U1t 2= L1t 2- E1t2

E1t 2

L1t 2

t 2001 2002 2003 2004 2005 2006 2007 2008

Labor force L(t)

(thousands)

143,734 144,863 146,510 147,401 149,320 151,428 153,124 154,287

Employed E(t)

(thousands)

136,933 136,485 137,736 139,252 141,730 144,427 146,047 145,362

t 1970 1980 1990 2000 2007

Smokers S(t) (thousands) 76,035 75,212 63,421 65,571 62,737

Population P(t) (thousands) 203,302 226,542 248,710 281,422 301,621

SECTION 6.2

■

Power Functions: Positive Powers 493

42. Crude Oil Imports The table shows the annual consumption as well as the annual

imports of crude oil and petroleum products from 2002 to 2008, as reported by the

Energy Information Administration of the U.S. Department of Energy.

(a) What is the meaning of the quotient function ?

(b) Calculate the values of for the indicated years. What do you conclude? R1t 2

R1t 2=

I1t 2

C1t 2

t 2002 2003 2004 2005 2006 2007 2008

Barrels imported per day

I(t) (thousands)

10,546 11,238 12,097 12,549 12,390 12,036 11,113

Barrels consumed per day

C(t) (thousands)

19,761 20,034 20,731 20,802 20,687 20,680 19,498

6.2 Power Functions: Positive Powers

■

Power Functions with Positive Integer Powers

■

Direct Proportionality

■

Fractional Positive Powers

■

Modeling with Power Functions

IN THIS SECTION… we study power functions with positive powers, and their graphs. We

compare them with exponential functions and see how they arise in geometry and science.

GET READY… by reviewing the rules of exponents in Algebra Toolkits A.3 and A.4,

and solving power equations in Algebra Toolkit C.1. Test your understanding by doing

the Algebra Checkpoint at the end of this section.

Some of the most familiar examples of functions are power functions. In geometry the

formulas for the area of a square or a circle and the volume of a

cube or a sphere all involve power functions. In physics, power

functions arise in Galileo’s formula for the distance traveled by a falling

object and in Kepler’s Third Law for the period of revolution of a planet

around the sun. (See Exercise 16, Section 6.4.). We will also see that power functions

occur in biology, where they help us to determine the number of species that a habi-

tat can support or how fast an animal walks naturally.

1T = Cd

3>2

1d = 16t

1V =

21V = x

1A = pr

21A = x

■ Power Functions with Positive Integer Powers

A power function is a function of the form

where the constant C and the power p are any nonzero numbers.

f 1x2= Cx

A power function is one in which the variable is raised to a fixed power.

Power Functions