Notes 3-8
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![Page 1: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/1.jpg)
Section 3-8Inverse Functions
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Warm-upIndicate how you would “undo” each operation or
composite of operations.
1. Turn east and walk 50 meters, then turn north and walk 30 meters.
3. Add -70 to a number, then multiply the result by 14.
2. Multiply a number by .45
4. Square a positive number, then cube it.
![Page 3: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/3.jpg)
Inverse of a function:
![Page 4: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/4.jpg)
Inverse of a function:
A function that will “undo” what another function had previously done
![Page 5: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/5.jpg)
Inverse of a function:
A function that will “undo” what another function had previously done
When the independent variable is switched with the dependent variable
![Page 6: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/6.jpg)
Inverse of a function:
A function that will “undo” what another function had previously done
When the independent variable is switched with the dependent variable
**Notation: The inverse of f is f-1
![Page 7: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/7.jpg)
Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
b. Describe S and its inverse in words.
![Page 8: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/8.jpg)
Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}
b. Describe S and its inverse in words.
![Page 9: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/9.jpg)
Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}
b. Describe S and its inverse in words.
S is a squaring function, where the independent variable is squared to obtain the dependent variable.
![Page 10: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/10.jpg)
Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}
b. Describe S and its inverse in words.
S is a squaring function, where the independent variable is squared to obtain the dependent variable.Its inverse is a positive square root function, where you would square root the independent variable to get the dependent variable.
![Page 11: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/11.jpg)
Just a little note:
![Page 12: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/12.jpg)
Just a little note:
As the independent variable switches with the dependent variable, the domain switches with the range.
![Page 13: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/13.jpg)
Theorem(Horizontal-Line Test)
![Page 14: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/14.jpg)
Theorem(Horizontal-Line Test)
If you can draw a horizontal line on the graph of f and it touches the graph more than once, then the
INVERSE of f is not a function.
![Page 15: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/15.jpg)
Theorem(Horizontal-Line Test)
If you can draw a horizontal line on the graph of f and it touches the graph more than once, then the
INVERSE of f is not a function.
The horizontal-line test tells us nothing about the original function...remember that!
![Page 16: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/16.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
![Page 17: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/17.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
y = 6x + 5
![Page 18: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/18.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
y = 6x + 5
x = 6y + 5
![Page 19: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/19.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
y = 6x + 5
x = 6y + 5 −5 −5
![Page 20: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/20.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
y = 6x + 5
x = 6y + 5 −5 −5
x − 5 = 6y
![Page 21: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/21.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
y = 6x + 5
x = 6y + 5 −5 −5
x − 5 = 6y
y =
x − 56
![Page 22: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/22.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
y = 6x + 5
x = 6y + 5 −5 −5
x − 5 = 6y
y =
x − 56 or
![Page 23: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/23.jpg)
Example 2Give an equation for the inverse and tell whether it is
a function.a. f x( ) = 6x + 5
y = 6x + 5
x = 6y + 5 −5 −5
x − 5 = 6y
y =
x − 56 or
y =
16
x −56
![Page 24: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/24.jpg)
Example 2b.
y =
43x − 1
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Example 2b.
y =
43x − 1
x =
43y − 1
![Page 26: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/26.jpg)
Example 2b.
y =
43x − 1
x =
43y − 1
3y − 1 =
4x
![Page 27: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/27.jpg)
Example 2b.
y =
43x − 1
x =
43y − 1
3y − 1 =
4x
3y =
4x+ 1
![Page 28: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/28.jpg)
Example 2b.
y =
43x − 1
x =
43y − 1
3y − 1 =
4x
3y =
4x+ 1
![Page 29: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/29.jpg)
Example 2b.
y =
43x − 1
x =
43y − 1
3y − 1 =
4x
3y =
4x+ 1
3y =
4x+
xx
![Page 30: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/30.jpg)
Example 2b.
y =
43x − 1
x =
43y − 1
3y − 1 =
4x
3y =
4x+ 1
3y =
4x+
xx
![Page 31: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/31.jpg)
Example 2b.
y =
43x − 1
x =
43y − 1
3y − 1 =
4x
3y =
4x+ 1
3y =
4x+
xx
3y =4 + x
x
![Page 32: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/32.jpg)
Example 2b.
y =
43x − 1
x =
43y − 1
3y − 1 =
4x
3y =
4x+ 1
3y =
4x+
xx
3y =4 + x
x
y =
4 + x3x
![Page 33: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/33.jpg)
Question:How do you verify that two functions are inverses of
each other?
![Page 34: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/34.jpg)
Question:How do you verify that two functions are inverses of
each other?
Use the Inverse Function Theorem!
![Page 35: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/35.jpg)
Question:How do you verify that two functions are inverses of
each other?
Use the Inverse Function Theorem!
The IFT says that two functions f and g are inverses of each other IFF f(g(x)) = x for all x in the domain of
g AND g(f(x)) = x for all x in the domain of f.
![Page 36: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/36.jpg)
Example 3Verify that the functions in Example 2a are inverses of
each other.
![Page 37: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/37.jpg)
Example 3Verify that the functions in Example 2a are inverses of
each other.
To do this, we have to show that f(g(x)) = x and g(f(x)) = x.
![Page 38: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/38.jpg)
Example 3Verify that the functions in Example 2a are inverses of
each other.
To do this, we have to show that f(g(x)) = x and g(f(x)) = x.
Let’s calculate this together.
![Page 39: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/39.jpg)
Example 4Explain why the functions f and g, with f(m) = m2 and
g(m) = m-2 are not inverses.
![Page 40: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/40.jpg)
Example 4Explain why the functions f and g, with f(m) = m2 and
g(m) = m-2 are not inverses.
Calculate f(g(m)). If this composite does not give us a value of m, then we know they are not inverses. If it
does, then we have to check g(f(m)).
![Page 41: Notes 3-8](https://reader034.fdocuments.us/reader034/viewer/2022052523/5559c86fd8b42a93208b45fa/html5/thumbnails/41.jpg)
Homework
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Homework
p. 212 # 1 - 20
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