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
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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