Function Operations 8.5 8.5 1.Add or subtract functions. 2.Multiply functions. Composite Functions 12.1 1.Find the composition of two functions.

Function Operations 8.58.5

1. Add or subtract functions.2. Multiply functions.

Composite Functions12.112.11. Find the composition of two functions.

Add the following polynomials.

5x + 1 3x2 – 7x + 6

3x2 – 2x + 7

f(x) = 5x + 1

(f + g)(x) = = (5x + 1) + (3x2 – 7x + 6) = 3x2 – 2x + 7

g(x) = 3x2 – 7x + 6

f(x) + g(x)Always rewrite!!!

Copyright © 2011 Pearson Education, Inc.

Adding or Subtracting Functions

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

(f – g)(x) = f(x) – g(x).

f(x) = 3x + 1 g(x) = 5x + 2Find:

(f + g)(x) (f - g)(x)

(g - f)(x) (f - g)(-2)

= f(x) + g(x)

= (3x + 1) + (5x + 2)

= 8x + 3

= f(x) – g(x)

= (3x + 1) – (5x + 2)

= -2x – 1

= 3x + 1 – 5x – 2

= g(x) – f(x)

= (5x + 2) – (3x + 1)

= 2x + 1

= 5x + 2 – 3x – 1

= f(-2) – g(-2)

f(-2) = 3(-2) + 1 = -5

= (-5) – (-8 )

g(-2)= 5(-2) + 2 = -8

= 3

Always rewrite!!!

Slide 3- 5Copyright © 2011 Pearson Education, Inc.

Given f(x) = 4x – 1 and g(x) = 5x + 2, what is (f + g)(x)?

a) x + 4

b) x − 4

c) 9x + 1

d) 9x – 1

8.5

Slide 3- 6Copyright © 2011 Pearson Education, Inc.

Given f(x) = 4x – 1 and g(x) = 5x + 2, what is (f + g)(x)?

a) x + 4

b) x − 4

c) 9x + 1

d) 9x – 1

8.5

Multiplying Functions

(f g)(x) = f(x)∙g(x).

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f(x) = 2x + 7 and g(x) = x − 4

Find (f g)(x).

= (2x + 7)(x − 4)

= 2x2 − 8x + 7x – 28

= 2x2 − x – 28

(f g)(x) = f(x)∙g(x)

Always rewrite!!!

f(x) = – x2 – 8x + 2 g(x) = x + 2 h(x) = x – 8 Find:

(gh)(x) (fg)(0)

(fh)(-1) (f h)(x)

=g(x) ∙ h(x)

= (x + 2)(x – 8)

= x 2 – 6x - 16

= f(0) ∙ g(0)

= (2)(2)

= 4

= f(-1) ∙ h(-1)

= (9)(-9)

= -81

= f(x) ∙ h(x)

= (-x2 – 8x + 2)(x – 8)

= -x 3 + 66x – 16

f(-1) = -(-1) 2 – 8(-1) + 2 = -1 + 8 + 2 = 9

Slide 3- 10Copyright © 2011 Pearson Education, Inc.

Given f(x) = 3x – 2 and g(x) = 5x – 1, what is (f g)(x)?

a) 15x2 − 13x + 2

b) 15x2 − 13x − 2

c) 15x2 − 7x + 2

d) 15x2 − 7x − 2

8.5

Slide 3- 11Copyright © 2011 Pearson Education, Inc.

Given f(x) = 3x – 2 and g(x) = 5x – 1, what is (f • g)(x)?

a) 15x2 − 13x + 2

b) 15x2 − 13x − 2

c) 15x2 − 7x + 2

d) 15x2 − 7x − 2

8.5

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

f (2) =

f (a) =

f (x+4) =

f (g(x)) =

2(2) + 3 = 7

2a + 3

2(x + 4) + 3 =

Composition of Functions

(f ◦ g)(x) =

Nested FormatNested Format

2x + 8 + 3 = 2x + 11

Composition of Functions

f g x f g x

g f x g f x

Shorthand notation for substitution.Shorthand notation for substitution.

Nested FormatNested Format

Always rewrite composition of functions in nested format!Always rewrite composition of functions in nested format!

Read “f of g of x”.Read “f of g of x”.

If and find .( ) 3 8f x x ( ) 2 5,g x x 3f g

3 3f g f g

1f

3 81

11

Find f(1).

Simplify.

Substitute 1 for g(3)

Find g(3).

Write in nested format.

g(3) = 2(3) – 5 = 1

f(x) = x2 – 8x + 2 g(x) = x + 2 h(x) = x – 8 Find: 3hg xfh

xgf

= g(h(3))

h(3) = 3 – 8 = -5

= g(-5)

= -3

= h(f(x))

= h(x2 – 8x + 2)

= (x2 – 8x + 2) - 8

= f(g(x))

= f(x + 2)

= (x + 2)2 – 8(x + 2) + 2

= x2 – 8x – 6

= x2 + 4x + 4 – 8x – 16 + 2

= x2 – 4x – 10

(x + 2)2 (x + 2)(x + 2)x2 + 4x + 4

(x + 2)2 x2 + 4X

g(-5) = -5 + 2 = -3

Rewrite & Foil

Always rewrite!!!

Slide 12- 16Copyright © 2011 Pearson Education, Inc.

If f(x) = x + 7 and g(x) = 2x – 12, what is

a) 44

b) 3

c) 3

d) 44

4 .f g

Slide 12- 17Copyright © 2011 Pearson Education, Inc.

If f(x) = x + 7 and g(x) = 2x – 12, what is

a) 44

b) 3

c) 3

d) 44

4 .f g