Section 1.6

The Algebra of Functions

Algebra of Functions Part 1

• The operations of addition, subtraction, multiplication, and division can be used to form new functions from given functions.

Example 1 Combining Functions • Suppose a small company silk-screens T-shirts with slogans and

logos. The company has costs associated with making the T-shirts and earns revenue when T-shirts are sold. The cost and revenue functions are given by

where x represents the number of T-shirts made and sold. Suppose further that the company can make between 20 and 300 T-shirts each week.

a. Find the profit function .

b. Sketch a graph of all three functions on the same coordinate grid.

246 1.9C x x ( ) 215 0.041R x x x= -

P x R x C x

Combining Functions by Using Addition, Subtraction, Multiplication

and Division

x

y

2 4 6 8 10

2

4

6

8

10

0

y g x

y f g x

y f x

Combined Functions

Suppose f and g are functions, then the following are also functions:

• The sum function, f + g • The difference function, f - g• The product function, fg • The quotient function, f/g The domains of each of the new functions are the

input values common to both original functions. The quotient function has the additional restriction

that its domain does not contain the input values that make the denominator zero.

Example 2 Combining Functions Numerically

Suppose f and g are functions defined by the following table of values.

x -1 0 1 2 3 4

f(x) 6 5 4 3 2 Und.

g(x) Und. -1 0 3 8 15

Find the domains of f and g. Identify the domain of each of the following functions and construct a table of values for each: f + g, f – g, fg, f/g.

Example 3 Combining Functions Graphically

• Let f and g be defined by the following graphs. Assume that the domain of each is the set of real numbers.

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-5-4

-3-2

-1

1

23

45

0

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-5-4-3-2-1

12345

0

y g x y f x

Find the following values, if possible. If a value is undefined, state so and explain why. a. (f + g)(0) b. (g -f)(0)

Example 4 Combining Functions Symbolically

Let and ,

which are both defined for all real numbers.

Find the symbolic representation of following functions and state the domain of each.

a. f + g b. f – g c. fg d. f/g e. g/f

2 1f x x 1g x x

Applications of Combining Functions

• Suppose x is the number of items made. The following functions are found by using basic functions

• Total Costs: The total amount of money associated with making x items.

• Revenue: The total amount of money received from the sale of x items.

R(x) = (number of items sold)(price per item)• Profit: The amount that revenue exceeds total costs. Negative

profit is called loss P(x) = R(x) – C(x) Unit Cost: The cost per unit when x items are made. C(x)/x

• Unit Profit: The profit per unit when x items are sold. P(x)/x

Fixed costs + variable costsC x

Example 5 An Application of Combining Functions

• Suppose a manufacturer has fixed costs of $5000, variable costs of $4 for each unit made, and sells the product for $20. Write a function that represents the following.

• a. Total cost• b. Unit cost• c. Revenue• d. Profit• e. Unit profit

Composition of Functions

• Another way of combining functions is to use the output value of one function as the input value of a second function.

• For example, suppose you are taking a trip and you will be visiting two countries, Japan and then Mexico. You might use a function to convert your dollars to yen in Japan and then another function to convert your yen to pesos in Mexico. Our composite function would be a single function that would convert the input in dollars directly to pesos.

Dollars Yen Pesos

g f

f of g

Composition of Functions

• The new operation is called composition and it is written as , which is read “f composed with g.” The result of composition is a new function that yields the result of finding the first output, g(x), and then using that output as the input of f to find , so another way to write the composition of f with g is f[g(x)].

f g

f g

( )f g x

Example 6 Composition of Functions Numerically

x 0 1 2 3

f(x) 4 3 2 1

g(x) 1 2 0 3

• Use the table of values to find the indicated function values. If a value is undefined, explain why.

• a. f(g(2)) b. f(g(1)) c. f(g(3)) d. g(f(1)) • e. g(f(2)) f. g(f(2)) g. f(f(1)) h. g(g(-4))

Composition of Functions Graphically

• Composition of two functions can also be found by using graphs of the two functions to find values.

Example 7 Composition of Functions Graphically

• Suppose that f and g are defined by the graphs below.

• a.• b. • c. • d.• e.

(0)f g

(5)f g

(3)g f (0)g f

(4)f f (2)g f

Example 8 Applying Composition of Functions

Suppose a circular puddle is slowly evaporating and the radius of the puddle (in inches) is

t minutes after it stops raining. The area of the

puddle is given by . • Find a function that gives the area of the puddle

t minutes after it stops raining.• Find the area of the puddle 10 minutes after it

stops raining.

20

3 1r t

t

2A r r

Composition of Inverse Functions

• When both orderings of a relation are functions and inverse functions are composed, the result is the original input. That is, suppose and

and .

y f x 1x f y

1f f y y 1f f x x

Example 9 Composition of Inverse Functions

• Suppose y = f(x) = 3x + 4 and x = g(y) = y/3 + 4/3. Find the following:

• a. • b.• c. Determine whether the functions are inverses of each

other

( )f g x( )( )g f x