slide1
Download
Skip this Video
Download Presentation
9-4

Loading in 2 Seconds...

play fullscreen
1 / 37

9-4 - PowerPoint PPT Presentation


  • 106 Views
  • Uploaded on

Operations with Functions. 9-4. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra2. x – 3. x – 2. Warm Up Simplify. Assume that all expressions are defined. 1. (2 x + 5) – ( x 2 + 3 x – 2). – x 2 – x + 7. 2. ( x – 3)( x + 1) 2. x 3 – x 2 – 5 x – 3. 3.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' 9-4' - enya


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Operations with Functions

9-4

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra2

slide2

x – 3

x – 2

Warm Up

Simplify. Assume that all expressions are defined.

1. (2x + 5) – (x2 + 3x – 2)

–x2 – x + 7

2. (x – 3)(x+ 1)2

x3 – x2 – 5x – 3

3.

slide3

Objectives

Add, subtract, multiply, and divide functions.

Write and evaluate composite functions.

slide4

Vocabulary

composition of functions

slide5

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.

slide7

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.

slide8

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.

Distributive Property

= 4x2 + 3x – 1 – 6x – 2

Combine like terms.

= 4x2–3x – 3

slide9

Check It Out! Example 1a

Given f(x) = 5x –6and 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.

slide10

Check It Out! Example 1b

Given f(x) = 5x –6and 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.

Distributive Property

= 5x – 6– x2+ 5x – 6

Combine like terms.

= –x2+ 10x – 12

slide12

Example 2A: Multiplying and Dividing Functions

Given f(x) = 6x2– x –12and 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

= 12x3 – 18x2 – 2x2+ 3x– 24x + 36

Multiply.

= 12x3 – 20x2 – 21x + 36

Combine like terms.

slide13

( )

( )

f

f

3

3

(x)

(x)

g

g

2

2

f(x)

6x2 – x –12

=

g(x)

2x – 3

=

=

=

Factor completely. Note that x ≠ .

(2x – 3)(3x +4)

(2x – 3)(3x + 4)

2x – 3

(2x – 3)

= 3x + 4, where x ≠

Example 2B: Multiplying and Dividing Functions

Set up the division as a rational expression.

Divide out common factors.

Simplify.

slide14

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

slide15

g

f

( )

( )

g

g(x)

(x)

(x)

=

f

f(x)

x2 – 4

=

x + 2

=

=

(x – 2)(x + 2)

(x – 2)(x + 2)

x + 2

(x + 2)

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.

slide16

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

slide17

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.

slide18

Reading Math

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

slide19

Caution!

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

slide20

Example 3A: Evaluating Composite Functions

Given f(x) = 2xand 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)

f(3) = 23

f(x) = 2x

= 8

So f(g(4)) = 8.

slide21

Example 3B: Evaluating Composite Functions

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

g(f(4))

Step 1 Find f(4)

f(4) = 24

f(x) = 2x

= 16

Step 2 Find g(16)

g(16) = 7 – 16

g(x) = 7 – x.

= –9

So g(f(4)) = –9.

slide22

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

= 9

Step 2 Find f(9)

f(9) = 2(9) – 3

f(x) = 2x – 3

= 15

So f(g(3)) = 15.

slide23

Check It Out! Example 3b

Given f(x) = 2x –3and g(x) = x2, 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.

slide24

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.

slide25

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

x

1 – x

f(g(x)) = f( )

x

x

=

= ()2– 1

1 – x

1 – x

–1 + 2x

(1 – x)2

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 {x|x ≠ 1} because g(1) is undefined.

slide26

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

(x2 – 1)

x

=

1 – (x2 – 1)

1 – x

=

x2– 1

2 – x2

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

slide27

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

f(g(x)) = 3( + 2) – 4

= + 6 – 4

= + 2

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 {x|x ≥ 0}.

slide28

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

g(f(x)) =

4

4

4

3

3

3

=

Note that x ≥ .

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

Check It Out! Example 4b

g(f(x))

Substitute the rule f into g.

slide30

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.

slide31

D(c) =

c

11.30

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.

slide32

= 1.15 ( )

D(P(c) ) = 1.15 ( )

c

1800

11.30

11.30

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.

slide33

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

slide34

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.

slide35

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.

slide36

f

g

( )

(x)

2x + 1

Lesson Quiz: Part I

Given f(x) = 4x2– 1and 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))

29

slide37

Given f(x) = x2and g(x) = , write each composite function. State the domain of each.

Lesson Quiz: Part II

f(g(x)) = x – 1;

5. f(g(x))

{x|x ≥ 1}

6. g(f(x))

{x|x ≤ – 1 or x ≥ 1}

ad