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LESSON 1

Operations with Functions

1

Perform Operations with Functions

 You have performed arithmetic operations

with polynomials. You can also use addition, subtraction, multiplication, and division

with functions.

You can perform arithmetic operations according to the following rules.

Key Concept 

Operations on Functions

Operation

Definition

Example

Let

f

(

x

)

=

2

x

and

g

(

x

)

=-

x

+

5.

Addition

(

f

+

g

)(

x

)

=

f

(

x

)

+

g

(

x

)

2

x

+

(

-

x

+

5)

=

x

+

5

Subtraction

(

f

-

g

)(

x

)

=

f

(

x

)

-

g

(

x

)

2

x

-

(

-

x

+

5)

=

3

x

-

5

Multiplication

(

f

·

g

)(

x

)

=

f

(

x

)

·

g

(

x

)

2

x

(

-

x

+

5)

= -

2​

x​

2

+

10

x

Division

(

​ 

f 

_ 

g

)

(

x

)

=

​ 

f

(

x

)

_ 

g

(

x

)

​,

g

(

x

)

0

​ 

2

x 

_ 

-

x

+

5 

​,

x

5

Given

f

(

x

)

=

x

2

-

4 and

g

(

x

)

=

2

x

+

1, find each function.

a.

(

f

+

g

)(

x

)

(

f

+

g

)(

x

)

=

f

(

x

)

+

g

(

x

)

Addition of functions

=

(​

x​

2

-

4)

+

(2

x

+

1)

f 

(

x 

)

=

x​

2

-

4 and

g

(

x 

)

=

2

x

+

1

=

x​

2

+

2

x

-

3

Simplify.

b.

(

f

-

g

)(

x

)

(

f

-

g

)(

x

)

=

f

(

x

)

-

g

(

x

)

Subtraction of functions

=

(​

x​

2

-

4)

-

(2

x

+

1)

f 

(

x 

)

=

x​

2

-

4 and

g

(

x 

)

=

2

x

+

1

=

x​

2

-

2

x

-

5

Simplify.

Guided Practice

Given

f

(

x

)

=

x​

2

+

5

x

-

2 and

g

(

x

)

=

3

x

-

2, find each function.

1A.

(

f

+

g

)(

x

)

1B.

(

f

-

g

)(

x

)

Example 1

Add and Subtract Functions

Why?

The graphs model the income

for the Brooks family since

2005, where

m

(

x

) represents

Mr. Brooks’ income and

f

(

x

)

represents Mrs. Brooks’ income.

The total household income for

the Brooks household can be

represented by

f

(

x

)

+

m

(

x

).

Now

1

Perform arithmetic

operations with

functions.

2

Apply arithmetic

operations with

functions.

Then

You performed

operations on

polynomials. 

New

Vocabulary

composition of

functions

Mathematical

Practices

4 

Model with

mathematics.

7 

Look for and make use

of structure.

10,000

0

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

100,000

m(x)

+

f(x)

f(x)

m(x)

2

Years Since 2005

Income ($)

4 6 8 10

Brooks Household Income

C07-001A-88482-B

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