1. 6.4 - Use Inverse Functions
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omit 13
g(f(3)) = -25, f(g(1)) = 1, f(f(0)) = 4 17 30 21 25
f•(f)(x):1/x2
f(f)(x): x 29
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2. 6.4 - Use Inverse Functions
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3. section 6.4 Revised ©2013, vdang@houstonisd.org
Inverse Functions
section 6.4
Revised ©2013,
6.4 - Use Inverse Functions
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4. 6.4 - Use Inverse Functions
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)
Resolve for y.
You may need to do the “SAT trick” becomes
Written in function notation as f –1(x)
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5. 6.4 - Use Inverse Functions
Example 1
Determine the inverse of this relation, {(0, –3), (2, 1), and (6, 3)} …to find the inverse, switch the x’s and y’s
{(0, –3), (2, 1), (6, 3)}
{( , ), ( , ), ( , )} –3 2 1 6 3
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6. 6.4 - Use Inverse Functions
Example 1
inverse
Mirrored Image
y = x
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7. In f’(x), we have the HORIZONTAL LINE TEST
Question
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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8. 6.4 - Use Inverse Functions
Example 2
Determine the inverse of y = 3x – 2
…to find the inverse, switch the x’s and y’s
y = 3x – 2 y
= 3 – 2 x
x = 3y – 2
x + 2 = 3y
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9. 6.4 - Use Inverse Functions
Example 3
Determine the inverse of y = (1/4)x – (1/2)
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10. 6.4 - Use Inverse Functions
Your Turn
Determine the inverse of y = 5x + 1/3
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11. 6.4 - Use Inverse Functions
Example 3
Determine the inverse of y = 4x2
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12. 6.4 - Use Inverse Functions
Example 4
Determine the inverse of y = 3x2 – 5 if x > 0
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13. 6.4 - Use Inverse Functions
Your Turn
Determine the inverse of y = x2 + 2 if x < 0
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14. 6.4 - Use Inverse Functions
Example 4
Determine the inverse of f(x) = 2x and determine whether the inverse is a function.
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15. 6.4 - Use Inverse Functions
Example 4
Determine the inverse of f(x) = 2x and determine whether the inverse is a function.
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16. 6.4 - Use Inverse Functions
Example 5
Determine the inverse of and determine whether the inverse is a function
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17. 6.4 - Use Inverse Functions
Example 6
Determine the inverse of f(x) = x2 – 2 and determine whether the inverse is a function
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18. 6.4 - Use Inverse Functions
Your Turn
Determine the inverse of and determine whether the inverse is a function
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19. 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 and (g(f(x)) is
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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20. 6.4 - Use Inverse Functions
Example 6
Prove that and are inverses through a composition.
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21. 6.4 - Use Inverse Functions
Example 7
Prove that and are inverses through a composition.
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22. 6.4 - Use Inverse Functions
Your Turn
Prove that and are inverses through a composition.
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23. 6.4 - Use Inverse Functions
Assignment
Pg 442: 3-11 odd, odd, EOO (do not graph)
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