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Holt McDougal Algebra 2
6-5 Operations with Functions6-5 Operations with Functions
Holt Algebra2
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
6-5 Operations with Functions
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
Holt McDougal Algebra 2
6-5 Operations with Functions
Add, subtract, multiply, and divide functions. Write and evaluate composite functions.
Objectives
Holt McDougal Algebra 2
6-5 Operations with Functions
Holt McDougal Algebra 2
6-5 Operations with Functions
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
Holt McDougal Algebra 2
6-5 Operations with Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function.
Example 1B: 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 + 3x – 1 – 6x – 2 Distributive Property
= 4x2 – 3x – 3
Holt McDougal Algebra 2
6-5 Operations with Functions
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function.
(f + g)(x)
Substitute function rules.
(f + g)(x) = f(x) + g(x)
Combine like terms.
= (5x – 6) + (x2 – 5x + 6)
= x2
Check It Out! Example 1a
Holt McDougal Algebra 2
6-5 Operations with Functions
(f – g)(x)
Substitute function rules.
(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.
Holt McDougal Algebra 2
6-5 Operations with Functions
When you divide functions, be sure to note any domain restrictions that may arise.
Holt McDougal Algebra 2
6-5 Operations with Functions
Example 2A: Multiplying and Dividing Functions
(fg)(x)
Substitute function rules.
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
Holt McDougal Algebra 2
6-5 Operations with Functions
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
Holt McDougal Algebra 2
6-5 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 McDougal Algebra 2
6-5 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 McDougal Algebra 2
6-5 Operations with Functions
The composition (f o g)(x) or f(g(x)) is read “f of g of x.”
Reading Math
Holt McDougal Algebra 2
6-5 Operations with Functions
Be careful not to confuse the notation for multiplication of functions with composition fg(x) ≠ f(g(x))
Caution!
Holt McDougal Algebra 2
6-5 Operations with Functions
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
Holt McDougal Algebra 2
6-5 Operations with Functions
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.
Holt McDougal Algebra 2
6-5 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 McDougal Algebra 2
6-5 Operations with Functions
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.
Holt McDougal Algebra 2
6-5 Operations with Functions
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
Holt McDougal Algebra 2
6-5 Operations with Functions
Composite functions can be used to simplify a series of functions.
Holt McDougal Algebra 2
6-5 Operations with 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
Holt McDougal Algebra 2
6-5 Operations with Functions
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.
Holt McDougal Algebra 2
6-5 Operations with Functions
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.
Holt McDougal Algebra 2
6-5 Operations with Functions
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
Holt McDougal Algebra 2
6-5 Operations with Functions
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