1.8 inverse functions my domain is your range no! my range is your domain
TRANSCRIPT
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1.8 Inverse functions
My domain is your rangeNo! My range is your domain
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Input/Output table to show how this works
If g(x) = f -1(x) , then the range of f(x) is the domain of g(x).
x f(x) x g(x)1 9 9 12 8 8 2 3 7 7 3Can you think of the equations f(x) and
g(x)?
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f(x) = - x + 10g(x) = - x + 10 To check to see if they are inverses we
should use composition of functions.(f º g)(x) = - (- x + 10 ) + 10
= x – 10 + 10 so (f º g)(x) = x
(g º f)(x) = - (- x + 10 ) + 10Same problem so (g º f)(x) = x
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Lets try another set of inverse functions
f(x) = 3x + 2 g(x) = 1/3x – 2/3 (f º g)(x) = 3(1/3x – 2/3) + 2(f º g)(x) = x
(g º f)(x) = 1/3(3x + 2) – 2/3(g º f)(x) = x
What can you say about inverse functions?
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To check if 2 functions are inverse
Use the compositions of the two functions,
namely f(x) and g(x).
(f º g)(x) = x(g º f)(x) = x
The value will always be x
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Graphing inverse functions
h(x) = x2 and f(x) = √x
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The inverses reflects over the line y = x (Identity function)
h(x) = x2 and f(x) = √x
This graph does not look complete why?
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Horizontal Line Test
If you pass a horizontal line through the graph and it only touches in one point, then the function has an inverse.
Does not have an inverse, unless we limit the
domain
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Horizontal Line Test
Does this function have an inverse?
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One to One functions
For every dependent variable (output) there is only one independent variable
(input). Look at f(x) = 2x -3 If f(x) = 1, then x = 2If x = 2, then f(x) = 1
A function only has an inverse if and only if it is “one to one”.
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Would f(x) = x2 have an inverse?
If f(x) = 16, then x = 4 or – 4.
Which makes f(x) not one to one, so no inverse.
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How to find the Inverse
Given f(x) = 8x – 2, change f(x) to y y = 8x – 2
Interchange x and yx = 8y – 2
Solve for y. Add 2 and divide by 8
⅛x + ¼ = y ; so f -1(x) = ⅛x + ¼
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How would you check to make sure the functions are inverses?
f(f -1) = 8(⅛x + ¼) – 2 = x + 2 – 2 = x
f-1(f) = ⅛(8x – 2) + ¼ = x - ¼ + ¼ = x
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5 1)( xxf
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5 1)( xxf
5 1 yx
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5 1)( xxf
5 1 yx
15 yx
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5 1)( xxf
5 1 yx
15 yx
yx 15
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5 1)( xxf
5 1 yx
15 yx
yx 15
1)( 51 xxf
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Would it change if y was y2 ?
5 1)( xxf
5 1 yx
15 yx
yx 15
1)( 51 xxf
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Homework
Page 83 – 86 # 3, 9, 15, 23, 31, 43, 51, 63, 73,
91, 99
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Homework
Page 83 – 86 # 4, 11, 19, 29, 39, 47, 59, 69, 85,
95