1. Page 196 1) 3) 5) 7) 9)
11)
13)
g(f(3)) = -25, f(g(1)) = 1, f(f(0)) = 4
15) 1
17) 30
19)
f(g(x)) = (x + 3)2
g(f(x)) = x2 + 3
Domain: All Reals
21)
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3.7 - Inverse Functions

2. Revised ©2015 Pre-Calculus PreAP AB/Dual Viet.dang@humble.k12.tx.us
Inverse Functions
Revised ©2015
Pre-Calculus PreAP AB/Dual
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3.7 - Inverse Functions

3. Definitions
The result of exchanging the input and output value of a relation is an Inverse Function
An inverse “undoes” the function. It switches (x, y) to (y, x)
Interchange the x and the y. (make y x and make x y)
Written in function notation f –1(x)
All inverses must be:
F is one-to-one
Graph must pass the horizontal line test
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3.7 - Inverse Functions

4. {(0, –3), (2, 1), (6, 3)} {( , ), ( , ), ( , )} –3 2 1 6 3 Example 1
Determine the inverse of this relation, {(0, –3), (2, 1), and (6, 3)}
{(0, –3), (2, 1), (6, 3)}
…to find the inverse, switch the x’s and y’s
{( , ), ( , ), ( , )} –3 2 1 6 3
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3.7 - Inverse Functions

5. Example 1 Mirrored Image y = x inverse 5/15/2019 12:07 PM
3.7 - Inverse Functions

6. In f’(x), we have the HORIZONTAL LINE TEST
Example 1
If a function is a relation, is an inverse a function as well?
inverse NO
REPEATING
Y’s
In f’(x), we have the HORIZONTAL LINE TEST
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3.7 - Inverse Functions

7. y = 3x – 2 y = 3 – 2 x x = 3y – 2 x + 2 = 3y Example 2
Determine the inverse of y = 3x – 2
y = 3x – 2 y
= 3 – 2 x
x = 3y – 2
x + 2 = 3y
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3.7 - Inverse Functions

8. Example 3 Determine the inverse of 𝒇 𝒙 = 𝟓 𝒙−𝟐 5/15/2019 12:07 PM
3.7 - Inverse Functions

9. Your Turn Determine the inverse of 𝒇 𝒙 = 𝟒𝒙−𝟑 𝟐 5/15/2019 12:07 PM
3.7 - Inverse Functions

10. Example 4 Determine the inverse of y = 4x2 5/15/2019 12:07 PM
3.7 - Inverse Functions

11. Your Turn Determine the inverse of y = 3x2 – 5 5/15/2019 12:07 PM
3.7 - Inverse Functions

12. Example 5 Determine the inverse of 𝒚= 𝟓 𝟑𝒙−𝟏 𝒙−𝟐 5/15/2019 12:07 PM
3.7 - Inverse Functions

13. Example 6 Determine the inverse of 𝒚= 𝒙 𝟐 𝒙 𝟐 +𝟏 5/15/2019 12:07 PM
3.7 - Inverse Functions

14. Your Turn Determine the inverse of 𝒚= 𝒙+𝟏 𝒙−𝟐 5/15/2019 12:07 PM
3.7 - Inverse Functions

15. One-to-One Functions
One-to-One Functions are functions that each x has only one y-value and each y has only one x-value. For a function to be one-to-one, it has to pass both the vertical and HORIZONTAL line test.
If a function is one-to-one, its inverse is also a function.
HORIZONTAL LINE TEST determines if the function is one-to-one.
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3.7 - Inverse Functions

16. Example 7
Use a calculator to determine if the relation is a function, 𝒇 𝒙 = 𝒙 𝟒 −𝟒 𝒙 𝟐 +𝟑 . If so, is the inverse a function?
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3.7 - Inverse Functions

17. Your Turn
Use a calculator to determine if the relation is a function, 𝒇 𝒙 = 𝟕𝒙 𝟓 +𝟑 𝒙 𝟒 −𝟐 𝒙 𝟑 +𝟐𝒙+𝟏 . If so, is the inverse a function?
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3.7 - Inverse Functions

18. Verifying Using Compositions
Composite f(g(x)). Take the g(x) function and substitute this into the f-function and simplify. For g(f(x)) take f(x) function and substitute this into the g-function and simplify.
Notation f(g(x)) is also 𝒇∘𝒈 𝒙 and (g(f(x)) is 𝒈∘𝒇 𝒙
If 2 functions are inverses then f(g(x)) = x and g(f(x)) = x. Both equations have to equal x.
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3.7 - Inverse Functions

19. Example 8
Prove that f(x) = 2x – 6 and g(x) = ½ x + 3 are inverses through a composition.
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3.7 - Inverse Functions

20. Your Turn
Prove that 𝒇 𝒙 = 𝒙−𝟐 𝟓 and 𝒈 𝒙 =𝟓𝒙+𝟐 are inverses through a composition.
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3.7 - Inverse Functions

21. Assignment
Page 212: 1, 3, 9-29 EOO (25 and 29 use Graphing Calculator), 31 (identify the inverse only), odd
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3.7 - Inverse Functions