9-4

9-4 Operations
Operations with
withFunctions
Functions

Warm Up
Lesson Presentation
Lesson Quiz

Holt
Holt Algebra2
Algebra 2
 9-4 Operations with Functions

Warm Up
Simplify. Assume that all expressions
are defined.
2
1. (2x + 5) – (x2 + 3x – 2) –x – x + 7

2. (x – 3)(x + 1)2 x3 – x2 – 5x – 3

3. x–3
x–2

Holt Algebra 2
 9-4 Operations with Functions

Objectives
Add, subtract, multiply, and divide
functions.
Write and evaluate composite
functions.

Holt Algebra 2
 9-4 Operations with Functions

Vocabulary
composition of functions

Holt Algebra 2
 9-4 Operations with Functions

You can perform operations on functions in

much the same way that you perform
operations on numbers or expressions. You can
add, subtract, multiply, or divide functions by
operating on their rules.

Holt Algebra 2
 9-4 Operations with Functions

Holt Algebra 2
 9-4 Operations with Functions
Example 1A: Adding and Subtracting Functions

Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2,

find each function.
(f + g)(x)

(f + g)(x) = f(x) + g(x)

= (4x2 + 3x – 1) + (6x + 2) Substitute function rules.

= 4x2 + 9x + 1 Combine like terms.

Holt Algebra 2
 9-4 Operations with Functions
Example 1B: Adding and Subtracting Functions

Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2,

find each function.
(f – g)(x)

(f – g)(x) = f(x) – g(x)

= (4x2 + 3x – 1) – (6x + 2) Substitute function rules.

= 4x2 + 3x – 1 – 6x – 2 Distributive Property

= 4x2 – 3x – 3 Combine like terms.

Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 1a

Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6,

find each function.
(f + g)(x)

(f + g)(x) = f(x) + g(x)

= (5x – 6) + (x2 – 5x + 6) Substitute function rules.

= x2 Combine like terms.

Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 1b

Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6,

find each function.
(f – g)(x)

(f – g)(x) = f(x) – g(x)

= (5x – 6) – (x2 – 5x + 6) Substitute function rules.

2
= 5x – 6 – x + 5x – 6 Distributive Property

= –x2 + 10x – 12 Combine like terms.

Holt Algebra 2
 9-4 Operations with Functions

When you divide functions, be sure to note any

domain restrictions that may arise.

Holt Algebra 2
 9-4 Operations with Functions
Example 2A: Multiplying and Dividing Functions
2
Given f(x) = 6x – x – 12 and g(x) = 2x – 3,
find each function.
(fg)(x)
(fg)(x) = f(x) ● g(x)

= (6x2 – x – 12) (2x – 3) Substitute function

rules.
2
= 6x (2x – 3) – x(2x – 3) – 12(2x – 3) Distributive Property

= 12x3 – 18x2 – 2x2 + 3x – 24x + 36 Multiply.

= 12x3 – 20x2 – 21x + 36 Combine like terms.

Holt Algebra 2
 9-4 Operations with Functions
Example 2B: Multiplying and Dividing Functions

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

6x2 – x –12 Set up the division as a

=
2x – 3 rational expression.
(2x – 3)(3x + 4) Factor completely.
= 3
2x – 3 Note that x ≠ 2 .
(2x – 3)(3x +4) Divide out common
=
(2x – 3) factors.
3
= 3x + 4, where x ≠ 2 Simplify.
Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 2a

Given f(x) = x + 2 and g(x) = x2 – 4, find each

function.
(fg)(x)

(fg)(x) = f(x) ● g(x)

= (x + 2)(x2 – 4) Substitute function rules.

= x3 + 2x2 – 4x – 8 Multiply.

Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 2b
g
( ) f
(x)
g
( ) f (x) =
g(x)
f(x)
x2 – 4 Set up the division as a
= rational expression.
x+2
(x – 2)(x + 2) Factor completely.
=
x+2 Note that x ≠ –2.
(x – 2)(x + 2) Divide out common
=
(x + 2) factors.
= x – 2, where x ≠ –2 Simplify.
Holt Algebra 2
 9-4 Operations with Functions

Another function operation uses the output from

one function as the input for a second function.
This operation is called the composition of
functions.

Holt Algebra 2
 9-4 Operations with Functions

The order of function operations is the same as the

order of operations for numbers and expressions. To
find f(g(3)), evaluate g(3) first and then substitute the
result into f.

Holt Algebra 2
 9-4 Operations with Functions

Reading Math
The composition (f o g)(x) or f(g(x)) is read “f of
g of x.”

Holt Algebra 2
 9-4 Operations with Functions

Caution!
Be careful not to confuse the notation for
multiplication of functions with composition
fg(x) ≠ f(g(x))

Holt Algebra 2
 9-4 Operations with Functions
Example 3A: Evaluating Composite Functions
x
Given f(x) = 2 and g(x) = 7 – x, find each
value.

f(g(4))

Step 1 Find g(4)

g(4) = 7 – 4 g(x) = 7 – x
=3
Step 2 Find f(3)
x
f(3) = 23 f(x) = 2
=8
So f(g(4)) = 8.
Holt Algebra 2
 9-4 Operations with Functions
Example 3B: Evaluating Composite Functions
x
Given f(x) = 2 and g(x) = 7 – x, find each
value.

g(f(4))

Step 1 Find f(4)

f(4) = 24 f(x) = 2
x

= 16
Step 2 Find g(16)
g(16) = 7 – 16 g(x) = 7 – x.
= –9
So g(f(4)) = –9.
Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 3a
2
Given f(x) = 2x – 3 and g(x) = x , find each
value.

f(g(3))

Step 1 Find g(3)

g(3) = 32 g(x) = x
2

=9
Step 2 Find f(9)
f(9) = 2(9) – 3 f(x) = 2x – 3
= 15
So f(g(3)) = 15.
Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 3b
2
Given f(x) = 2x – 3 and g(x) = x , find each
value.

g(f(3))

Step 1 Find f(3)

f(3) = 2(3) – 3 f(x) = 2x – 3
=3
Step 2 Find g(3)
g(3) = 32 g(x) = x2
=9
So g(f(3)) = 9.
Holt Algebra 2
 9-4 Operations with Functions

You can use algebraic expressions as well as

numbers as inputs into functions. To find a rule
for f(g(x)), substitute the rule for g into f.

Holt Algebra 2
 9-4 Operations with Functions
Example 4A: Writing Composite Functions
2
x
Given f(x) = x – 1 and g(x) = 1 – x , write
each composite function. State the domain of
each.
f(g(x))
x
f(g(x)) = f( ) Substitute the rule g into f.
1–x
x
=(
2
) –1 Use the rule for f. Note that
1–x x ≠ 1.
–1 + 2x
= 2 Simplify.
(1 – x)

The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because

g(1) is undefined.
Holt Algebra 2
 9-4 Operations with Functions
Example 4B: Writing Composite Functions
2
x
Given f(x) = x – 1 and g(x) = 1 – x , write
each composite function. State the domain of
each.
g(f(x))
g(f(x)) = g(x2 – 1) Substitute the rule f into g.
(x2 – 1)
= Use the rule for g.
2
1 – (x – 1)
x2 – 1
= Simplify. Note that x ≠ .
2 – x2
The domain of g(f(x)) is x ≠ or {x|x ≠ }
because f( ) = 1 and g(1) is undefined.
Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 4a

Given f(x) = 3x – 4 and g(x) = + 2 , write

each composite. State the domain of each.
f(g(x))

f(g(x)) = 3( + 2) – 4 Substitute the rule g into f.

= +6–4 Distribute. Note that x ≥ 0.

= +2 Simplify.

The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.

Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 4b

Given f(x) = 3x – 4 and g(x) = + 2 , write

each composite. State the domain of each.
g(f(x))

33x x 4 4 2  2
g(f(x)) = Substitute the rule f into g.

4
= Note that x ≥ 3 .

4 4
The domain of g(f(x)) is x ≥ 3 or {x|x ≥ 3 }.

Holt Algebra 2
 9-4 Operations with Functions

Composite functions can be used to simplify a

series of functions.

Holt Algebra 2
 9-4 Operations with Functions
Example 5: Business Application

Jake imports furniture from Mexico. The

exchange rate is 11.30 pesos per U.S. dollar.
The cost of each piece of furniture is given in
pesos. The total cost of each piece of furniture
includes a 15% service charge.

A. Write a composite function to represent the total

cost of a piece of furniture in dollars if the cost of
the item is c pesos.

Holt Algebra 2
 9-4 Operations with Functions
Example 5 Continued

Step 1 Write a function for the total cost in U.S.

dollars.
P(c) = c + 0.15c
= 1.15c

Step 2 Write a function for the cost in dollars based

on the cost in pesos.
c
D(c) = Use the exchange rate.
11.30

Holt Algebra 2
 9-4 Operations with Functions
Example 5 Continued

Step 3 Find the composition D(P(c)).

D(P(c)) = 1.15P(c) Substitute P(c) for c.
c
= 1.15 ( ) Replace P(c) with its rule.
11.30

B. Find the total cost of a table in dollars if it costs

1800 pesos.
Evaluate the composite function for c = 1800.
1800
D(P(c) ) = 1.15 ( )
11.30
≈ 183.19
The table would cost $183.19, including all charges.
Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 5

During a sale, a music store is selling all drum

kits for 20% off. Preferred customers also
receive an additional 15% off.
a. Write a composite function to represent the final
cost of a kit for a preferred customer that originally
cost c dollars.

Step 1 Write a function for the final cost of a kit that

originally cost c dollars.
f(c) = 0.80c Drum kits are sold at
80% of their cost.

Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 5 Continued

Step 2 Write a function for the final cost if the

customer is a preferred customer.

g(c) = 0.85c Preferred customers receive 15% off.

Holt Algebra 2
 9-4 Operations with Functions
Check It Out! Example 5 Continued
Step 3 Find the composition f(g(c)).
f(g(c)) = 0.80(g(c)) Substitute g(c) for c.

f(g(c)) = 0.80(0.85c) Replace g(c) with its rule.

= 0.68c
b. Find the cost of a drum kit at $248 that a preferred
customer wants to buy.
Evaluate the composite function for c = 248.
f(g(c) ) = 0.68(248)

The drum kit would cost $168.64.

Holt Algebra 2
 9-4 Operations with Functions
Lesson Quiz: Part I
2
Given f(x) = 4x – 1 and g(x) = 2x – 1, find
each function or value.

1. (f + g)(x) 4x2 + 2x – 2

2. (fg)(x) 8x3 – 4x2 – 2x + 1

f
3. ( )(x)
g 2x + 1

4. g(f(2)) 29

Holt Algebra 2
 9-4 Operations with Functions
Lesson Quiz: Part II
2
Given f(x) = x and g(x) = , write each
composite function. State the domain of each.

5. f(g(x)) f(g(x)) = x – 1;
{x|x ≥ 1}

6. g(f(x))
{x|x ≤ – 1 or x ≥ 1}

Holt Algebra 2