Transcript Document

8.5
Function Operations
1. Add or subtract functions.
2. Multiply functions.
12.1 Composite Functions
1. Find the composition of two functions.
Add the following polynomials.
5x + 1
3x2 – 7x + 6
3x2 – 2x + 7
f(x) = 5x + 1
g(x) = 3x2 – 7x + 6
Always rewrite!!!
(f + g)(x) = f(x) + g(x)
= (5x + 1) + (3x2 – 7x + 6)
= 3x2 – 2x + 7
Adding or Subtracting Functions
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x).
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Find:
f(x) = 3x + 1
Always rewrite!!!
(f + g)(x)
g(x) = 5x + 2
(f - g)(x)
= f(x) – g(x)
= f(x) + g(x)
= (3x + 1) – (5x + 2)
= (3x + 1) + (5x + 2)
= 3x + 1 – 5x – 2
= 8x + 3
(g - f)(x)
= -2x – 1
(f - g)(-2)
= (-5) – (-8 )
= g(x) – f(x)
= (5x + 2) – (3x + 1)
= 5x + 2 – 3x – 1
= 2x + 1
= f(-2) – g(-2)
=3
f(-2) = 3(-2) + 1 = -5
g(-2)= 5(-2) + 2 = -8
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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Slide 3- 5
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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Slide 3- 6
Multiplying Functions
(f g)(x) = f(x)∙g(x).
f(x) = 2x + 7 and g(x) = x − 4
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Find (f g)(x).
Always rewrite!!!
(f g)(x) = f(x)∙g(x)
= (2x + 7)(x − 4)
= 2x2 − 8x + 7x – 28
= 2x2 − x – 28
f(x) = – x2 – 8x + 2
Find:
(gh)(x)
(fh)(-1)
=g(x) ∙ h(x)
g(x) = x + 2
(fg)(0)
h(x) = x – 8
= f(0) ∙ g(0)
= (x + 2)(x – 8)
= (2)(2)
= x 2 – 6x - 16
=4
= f(-1) ∙ h(-1)
(f h)(x)
= f(x) ∙ h(x)
= (9)(-9)
= (-x2 – 8x + 2)(x – 8)
= -81
= -x 3 + 66x – 16
f(-1) = -(-1) 2 – 8(-1) + 2
= -1 + 8 + 2
=9
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
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Slide 3- 10
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
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Slide 3- 11
f(x) = 2x + 3
f (2) =
2(2) + 3 = 7
f (a) =
2a + 3
f (x+4) =
g(x) = x + 4
2(x + 4) + 3 = 2x + 8 + 3 = 2x + 11
f (g(x)) = Nested Format
(f ◦ g)(x) =
Composition of Functions
Composition of Functions
f
g
Read “f of g of x”.
g   x   f  g  x  
f
x 
g  f  x  
Nested Format
Shorthand notation for substitution.
Always rewrite composition of functions in nested format!
g(3) = 2(3) – 5 = 1
If f ( x )  3 x  8 and g ( x )  2 x  5, find  f

f
g 3  f
 g 3
g 3 .
Write in nested format.
Find g(3).
 f 1 
Substitute 1 for g(3)
 3 1   8
Find f(1).
 11
Simplify.
f(x) = x2 – 8x + 2
g(x) = x + 2
Always rewrite!!!
Find:
 g  h 3 
= g(h(3))
 h  f  x 
h(3) = 3 – 8 = -5
= h(f(x))
= g(-5)
= h(x2 – 8x + 2)
= -3
= (x2 – 8x + 2) - 8
g(-5) = -5 + 2 = -3
 f  g  x 
h(x) = x – 8
= f(g(x))
= f(x + 2)
= x2 – 8x – 6
(x + 2)2
(x + 2)(x + 2)
x2 + 4x + 4
= (x + 2)2 – 8(x + 2) + 2
= x2 + 4x + 4 – 8x – 16 + 2
= x2 – 4x – 10
Rewrite & Foil
X
(x + 2)2
x2 + 4
If f(x) = x + 7 and g(x) = 2x – 12, what
is  f g   4  .
a) 44
b) 3
c) 3
d) 44
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Slide 12- 16
If f(x) = x + 7 and g(x) = 2x – 12, what
is  f g   4  .
a) 44
b) 3
c) 3
d) 44
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Slide 12- 17