holt mcdougal algebra 2 6-5 operations with functions 6-5 operations with functions holt algebra2...

Holt McDougal Algebra 2

6-5 Operations with Functions6-5 Operations with Functions

Holt Algebra2

Warm Up

Lesson Presentation

Lesson Quiz



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



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

Objectives



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



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.



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



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.



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

holt mcdougal algebra 2 6-5 operations with functions 6-5 operations with functions holt algebra2...

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