Operations with Functions

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Operations with Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra2
Holt McDougal Algebra 2
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Warm Up Simplify. Assume that all expressions are
defined.
x2 x 7
1. (2x 5) (x2 3x 2)
2. (x 3)(x 1)2
x3 x2 5x 3
3.
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Objectives
Add, subtract, multiply, and divide functions.
Write and evaluate composite functions.
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Vocabulary
composition of functions
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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.
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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)
Substitute function rules.
(4x2 3x 1) (6x 2)
4x2 9x 1
Combine like terms.
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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)
Substitute function rules.
(4x2 3x 1) (6x 2)
Distributive Property
4x2 3x 1 6x 2
Combine like terms.
4x2 3x 3
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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)
Substitute function rules.
(5x 6) (x2 5x 6)
x2
Combine like terms.
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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)
Substitute function rules.
(5x 6) (x2 5x 6)
Distributive Property
5x 6 x2 5x 6
Combine like terms.
x2 10x 12
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When you divide functions, be sure to note any
domain restrictions that may arise.
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Example 2A Multiplying and Dividing Functions
Given f(x) 6x2 x 12 and g(x) 2x 3, find
each function.
(fg)(x)
(fg)(x) f(x) ? g(x)
Substitute function rules.
(6x2 x 12) (2x 3)
6x2 (2x 3) x(2x 3) 12(2x 3)
Distributive Property
Multiply.
12x3 18x2 2x2 3x 24x 36
Combine like terms.
12x3 20x2 21x 36
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Example 2B Multiplying and Dividing Functions
Set up the division as a rational expression.
Divide out common factors.
Simplify.
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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.
Multiply.
x3 2x2 4x 8
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Check It Out! Example 2b
Set up the division as a rational expression.
Factor completely. Note that x ? 2.
Divide out common factors.
x 2, where x ? 2
Simplify.
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Another function operation uses the output from
one function as the input for a second function.
This operation is called the composition of
functions.
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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.
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Example 3A Evaluating Composite Functions
Given f(x) 2x and g(x) 7 x, find each value.
f(g(4))
Step 1 Find g(4)
g(4) 7 4
g(x) 7 x
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Step 2 Find f(3)
f(3) 23
f(x) 2x
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So f(g(4)) 8.
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Example 3B Evaluating Composite Functions
Given f(x) 2x and g(x) 7 x, find each value.
g(f(4))
Step 1 Find f(4)
f(4) 24
f(x) 2x
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Step 2 Find g(16)
g(16) 7 16
g(x) 7 x.
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So g(f(4)) 9.
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Check It Out! Example 3a
Given f(x) 2x 3 and g(x) x2, find each
value.
f(g(3))
Step 1 Find g(3)
g(3) 32
g(x) x2
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Step 2 Find f(9)
f(9) 2(9) 3
f(x) 2x 3
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So f(g(3)) 15.
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Check It Out! Example 3b
Given f(x) 2x 3 and g(x) x2, find each
value.
g(f(3))
Step 1 Find f(3)
f(3) 2(3) 3
f(x) 2x 3
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Step 2 Find g(3)
g(3) 32
g(x) x2
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So g(f(3)) 9.
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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.
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Example 4A Writing Composite Functions
f(g(x))
Substitute the rule g into f.
Use the rule for f. Note that x ? 1.
Simplify.
The domain of f(g(x)) is x ? 1 or xx ? 1
because g(1) is undefined.
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Example 4B Writing Composite Functions
g(f(x))
g(f(x)) g(x2 1)
Substitute the rule f into g.
Use the rule for g.
Simplify. Note that x ? .
The domain of g(f(x)) is x ? or xx ?
because f( ) 1 and g(1) is undefined.
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Check It Out! Example 4a
f(g(x))
Substitute the rule g into f.
Distribute. Note that x 0.
Simplify.
The domain of f(g(x)) is x 0 or xx 0.
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Check It Out! Example 4b
g(f(x))
Substitute the rule f into g.
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Composite functions can be used to simplify a
series of functions.
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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.
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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.
Use the exchange rate.
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Example 5 Continued
Step 3 Find the composition D(P(c)).
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.
183.19
The table would cost 183.19, including all
charges.
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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.
Drum kits are sold at 80 of their cost.
f(c) 0.80c
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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.
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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.
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Lesson Quiz Part I
Given f(x) 4x2 1 and g(x) 2x 1, find each
function or value.
4x2 2x 2
1. (f g)(x)
2. (fg)(x)
8x3 4x2 2x 1
3.
4. g(f(2))
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Lesson Quiz Part II
f(g(x)) x 1
5. f(g(x))
xx 1
6. g(f(x))
xx 1 or x 1