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9-4 Operations with Functions

Holt Algebra2

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Lesson QuizLesson Quiz

Warm UpSimplify. Assume that all expressions are defined.

–x2 – x + 71. (2x + 5) – (x2 + 3x – 2)

2. (x – 3)(x + 1)2

3.

x3 – x2 – 5x – 3

x – 3 x – 2

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

Objectives

composition of functions

Vocabulary

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.

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

Example 1A: Adding and Subtracting Functions

(f + g)(x)

Substitute function rules.

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

Combine like terms.

= (4x2 + 3x – 1) + (6x + 2)

= 4x2 + 9x + 1

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

Example 1B: Adding and Subtracting Functions

(f – g)(x)


(f – g)(x) = f(x) – g(x)

Combine like terms.

= (4x2 + 3x – 1) – (6x + 2)

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

= 4x2 – 3x – 3

Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function.

(f + g)(x)


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

Combine like terms.

= (5x – 6) + (x2 – 5x + 6)

= x2

Check It Out! Example 1a

(f – g)(x)


(f – g)(x) = f(x) – g(x)

Combine like terms.

= (5x – 6) – (x2 – 5x + 6)

= 5x – 6 – x2 + 5x – 6 Distributive Property

= –x2 + 10x – 12

Check It Out! Example 1b

Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function.

When you divide functions, be sure to note any domain restrictions that may arise.

Example 2A: Multiplying and Dividing Functions

(fg)(x)


Multiply.

Combine like terms.

Distributive Property

= (6x2 – x – 12) (2x – 3)

Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3, find each function.

(fg)(x) = f(x) ● g(x)

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

= 12x3 – 18x2 – 2x2 + 3x – 24x + 36

= 12x3 – 20x2 – 21x + 36

Set up the division as a rational expression.

Divide out common factors.

Simplify.

( )(x)fg

( )(x)fg

f(x)g(x)=

6x2 – x –122x – 3

=

Factor completely. Note that x ≠ .3

2=

(2x – 3)(3x + 4)2x – 3

=(2x – 3)(3x +4)

(2x – 3)

= 3x + 4, where x ≠ 32

Example 2B: Multiplying and Dividing Functions

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)

= x3 + 2x2 – 4x – 8 Multiply.


Set up the division as a rational expression.

Divide out common factors.

Simplify.

( )(x)gf

( )(x)gf

g(x)f(x)

=

Factor completely. Note that x ≠ –2.

=(x – 2)(x + 2)

x + 2

=(x – 2)(x + 2)

(x + 2)

= x – 2, where x ≠ –2


=x2 – 4x + 2

Another function operation uses the output from one function as the input for a second function. This operation is called the composition of 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.

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

Reading Math

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

Caution!

Example 3A: Evaluating Composite Functions

Step 1 Find g(4)

g(x) = 7 – x

Given f(x) = 2x and g(x) = 7 – x, find each value.

f(g(4))

g(4) = 7 – 4

Step 2 Find f(3) = 3

f(3) = 23

= 8

So f(g(4)) = 8.

f(x) = 2x

Example 3B: Evaluating Composite Functions

Step 1 Find f(4)

f(x) = 2x

Given f(x) = 2x and g(x) = 7 – x, find each value.

g(f(4))

f(4) = 24

Step 2 Find g(16)

= 16

g(16) = 7 – 16= –9

So g(f(4)) = –9.

g(x) = 7 – x.

Step 1 Find g(3)

g(x) = x2

Given f(x) = 2x – 3 and g(x) = x2, find each value.

f(g(3))

g(3) = 32

Step 2 Find f(9) = 9

f(9) = 2(9) – 3= 15

So f(g(3)) = 15.

f(x) = 2x – 3


Step 1 Find f(3)

f(x) = 2x – 3

Given f(x) = 2x – 3 and g(x) = x2, find each value.

g(f(3))

f(3) = 2(3) – 3

Step 2 Find g(3) = 3

g(3) = 32

= 9

So g(f(3)) = 9.

g(x) = x2


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.

Example 4A: Writing Composite Functions

Given f(x) = x2 – 1 and g(x) = , write each composite function. State the domain of each.

x1 – x

f(g(x))

f(g(x)) = f( ) x1 – x

–1 + 2x

(1 – x)2=

= ( )2 – 1 x1 – x

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

Use the rule for f. Note that x ≠ 1.

Substitute the rule g into f.

Simplify.

Simplify. Note that x ≠ .

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

g(f(x)) = g(x2 – 1)

Example 4B: Writing Composite Functions

g(f(x))

Use the rule for g.

Substitute the rule f into g.

= x2 – 1

2 – x2

(x2 – 1)

1 – (x2 – 1)=

Given f(x) = x2 – 1 and g(x) = , write each composite function. State the domain of each.

x1 – x

f(g(x))

Distribute. Note that x ≥ 0.

Substitute the rule g into f.

Simplify.


Given f(x) = 3x – 4 and g(x) = + 2 , write each composite. State the domain of each.

f(g(x)) = 3( + 2) – 4

= + 6 – 4

= + 2

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

g(f(x))

Substitute the rule f into g.


g(f(x)) = 3 4 2x

3 4 2x = Note that x ≥ .43

The domain of g(f(x)) is x ≥ or {x|x ≥ }.43

43

Given f(x) = 3x – 4 and g(x) = + 2 , write each composite. State the domain of each.

Composite functions can be used to simplify a series of functions.

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.

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.

Example 5: Business Application

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

Example 5 Continued

P(c) = c + 0.15c

= 1.15c

Step 2 Write a function for the cost in dollars based on the cost in pesos.

D(c) = c11.30

Use the exchange rate.

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

Example 5 Continued

= 1.15 ( ) c11.30

D(P(c)) = 1.15P(c) Substitute P(c) for c.

Replace P(c) with its rule.

B. Find the total cost of a table in dollars if it costs 1800 pesos.

Evaluate the composite function for c = 1800.

D(P(c) ) = 1.15 ( )1800

11.30≈ 183.19

The table would cost $183.19, including all charges.

a. Write a composite function to represent the final cost of a kit for a preferred customer that originally cost c dollars.

During a sale, a music store is selling all drum kits for 20% off. Preferred customers also receive an additional 15% off.

Check It Out! Example 5

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.

Check It Out! Example 5 Continued

Step 2 Write a function for the final cost if the customer is a preferred customer.

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

Step 3 Find the composition f(g(c)).

f(g(c)) = 0.80(g(c)) Substitute g(c) for c.

Replace g(c) with its rule.

Check It Out! Example 5 Continued

f(g(c)) = 0.80(0.85c)

= 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.

Lesson Quiz: Part I

Given f(x) = 4x2 – 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

3. ( )(x)fg

4. g(f(2)) 29

2x + 1

Lesson Quiz: Part II

5. f(g(x)) f(g(x)) = x – 1;

{x|x ≤ – 1 or x ≥ 1}

Given f(x) = x2 and g(x) = , write each composite function. State the domain of each.

6. g(f(x))

{x|x ≥ 1}

2 1;g f x x