section 1.6 inverses - dr. travers page of...
TRANSCRIPT
![Page 1: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/1.jpg)
Section 1.6 Inverses
![Page 2: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/2.jpg)
What Is A Function?
Who can tell me what is a function?
DefinitionFor nonempty sets A and B, a function f maps A to B denotedf : A→ B, such that for each element a ∈ A, there is exactly oneelement b ∈ B. We write this as f (a) = b of b is the unique element ofB that is assigned to a ∈ A.
What functions can you name?
![Page 3: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/3.jpg)
What Is A Function?
Who can tell me what is a function?
DefinitionFor nonempty sets A and B, a function f maps A to B denotedf : A→ B, such that for each element a ∈ A, there is exactly oneelement b ∈ B. We write this as f (a) = b of b is the unique element ofB that is assigned to a ∈ A.
What functions can you name?
![Page 4: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/4.jpg)
What Is A Function?
Who can tell me what is a function?
DefinitionFor nonempty sets A and B, a function f maps A to B denotedf : A→ B, such that for each element a ∈ A, there is exactly oneelement b ∈ B. We write this as f (a) = b of b is the unique element ofB that is assigned to a ∈ A.
What functions can you name?
![Page 5: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/5.jpg)
Important Terms
Let f : X → Y be a function mapping X to Y .
DefinitionX is the domain of f .
![Page 6: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/6.jpg)
Important Terms
Let f : X → Y be a function mapping X to Y .
DefinitionX is the domain of f .
![Page 7: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/7.jpg)
Important Terms
What is the Y? (blue region here)
DefinitionY is the codomain of f .
![Page 8: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/8.jpg)
Important Terms
What is the Y? (blue region here)
DefinitionY is the codomain of f .
![Page 9: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/9.jpg)
Important Terms
So what is the yellow region here?
DefinitionThe range of f is the set of all f (x) where x ∈ X.
![Page 10: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/10.jpg)
Important Terms
So what is the yellow region here?
DefinitionThe range of f is the set of all f (x) where x ∈ X.
![Page 11: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/11.jpg)
In Terms of the Elements
Anyone know what we call x and y here?
Definitiony = f (x) is called the image of x.
So, the range is the set of all images of X.
![Page 12: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/12.jpg)
In Terms of the Elements
Anyone know what we call x and y here?
Definitiony = f (x) is called the image of x.
So, the range is the set of all images of X.
![Page 13: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/13.jpg)
In Terms of the Elements
Anyone know what we call x and y here?
Definitiony = f (x) is called the image of x.
So, the range is the set of all images of X.
![Page 14: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/14.jpg)
In Terms of the Elements
Definitionx is called the preimage of y = f (x).
![Page 15: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/15.jpg)
Why We Bring This Up
A function can only have an inverse if it is 1-1. Do we rememberwhat 1-1 means?
![Page 16: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/16.jpg)
Inverses
Definition
If f (a) = b, then the function f−1 is the inverse function of f andf−1(b) = a.
So what we are saying is that we need to not only have a uniqueoutput for each input (function) but we need to also have a uniqueinput for each output.
![Page 17: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/17.jpg)
Inverses
Definition
If f (a) = b, then the function f−1 is the inverse function of f andf−1(b) = a.
So what we are saying is that we need to not only have a uniqueoutput for each input (function) but we need to also have a uniqueinput for each output.
![Page 18: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/18.jpg)
Inverses Visually
x
y
Notice that the domain of the original function equals the range of theinverse and the domain of the inverse equals the range of the originalfunction ...
![Page 19: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/19.jpg)
Inverses Visually
x
y
Notice that the domain of the original function equals the range of theinverse and the domain of the inverse equals the range of the originalfunction ...
![Page 20: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/20.jpg)
Inverses from Tables
ExampleGiven the following table,
x 2 4 6 8f (x) 8 6 2 4
Find f (6) and f−1(4)
f (6) = 2
f−1(4) = 8
![Page 21: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/21.jpg)
Inverses from Tables
ExampleGiven the following table,
x 2 4 6 8f (x) 8 6 2 4
Find f (6) and f−1(4)
f (6) =
2
f−1(4) = 8
![Page 22: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/22.jpg)
Inverses from Tables
ExampleGiven the following table,
x 2 4 6 8f (x) 8 6 2 4
Find f (6) and f−1(4)
f (6) = 2
f−1(4) = 8
![Page 23: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/23.jpg)
Inverses from Tables
ExampleGiven the following table,
x 2 4 6 8f (x) 8 6 2 4
Find f (6) and f−1(4)
f (6) = 2
f−1(4) =
8
![Page 24: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/24.jpg)
Inverses from Tables
ExampleGiven the following table,
x 2 4 6 8f (x) 8 6 2 4
Find f (6) and f−1(4)
f (6) = 2
f−1(4) = 8
![Page 25: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/25.jpg)
Inverses from Graphs
ExampleFor the given graph,
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-2
-1
f (x)
find f (2) and f−1(4)
f (2) = 1f−1(4) = -1 and 3
![Page 26: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/26.jpg)
Inverses from Graphs
ExampleFor the given graph,
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-2
-1
f (x)
find f (2) and f−1(4)
f (2) =
1f−1(4) = -1 and 3
![Page 27: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/27.jpg)
Inverses from Graphs
ExampleFor the given graph,
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-2
-1
f (x)
find f (2) and f−1(4)
f (2) = 1
f−1(4) = -1 and 3
![Page 28: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/28.jpg)
Inverses from Graphs
ExampleFor the given graph,
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-2
-1
f (x)
find f (2) and f−1(4)
f (2) = 1f−1(4) =
-1 and 3
![Page 29: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/29.jpg)
Inverses from Graphs
ExampleFor the given graph,
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-2
-1
f (x)
find f (2) and f−1(4)
f (2) = 1f−1(4) = -1 and 3
![Page 30: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/30.jpg)
Finding Inverses
This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.
Example
Find the inverse of f (x) = 3x + 2.
1 Rewrite as ‘y =’y = 3x + 2
2 Switch x and yx = 3y + 2
3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2
3 = y4 Rewrite as ‘f−1(x) =’
f−1(x) = x−23
![Page 31: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/31.jpg)
Finding Inverses
This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.
Example
Find the inverse of f (x) = 3x + 2.
1 Rewrite as ‘y =’y = 3x + 2
2 Switch x and yx = 3y + 2
3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2
3 = y4 Rewrite as ‘f−1(x) =’
f−1(x) = x−23
![Page 32: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/32.jpg)
Finding Inverses
This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.
Example
Find the inverse of f (x) = 3x + 2.
1 Rewrite as ‘y =’y = 3x + 2
2 Switch x and yx = 3y + 2
3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2
3 = y4 Rewrite as ‘f−1(x) =’
f−1(x) = x−23
![Page 33: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/33.jpg)
Finding Inverses
This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.
Example
Find the inverse of f (x) = 3x + 2.
1 Rewrite as ‘y =’y = 3x + 2
2 Switch x and yx = 3y + 2
3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2
3 = y4 Rewrite as ‘f−1(x) =’
f−1(x) = x−23
![Page 34: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/34.jpg)
Finding Inverses
This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.
Example
Find the inverse of f (x) = 3x + 2.
1 Rewrite as ‘y =’y = 3x + 2
2 Switch x and yx = 3y + 2
3 Solve for y
x = 3y + 2⇒ x− 2 = 3y⇒ x−23 = y
4 Rewrite as ‘f−1(x) =’f−1(x) = x−2
3
![Page 35: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/35.jpg)
Finding Inverses
This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.
Example
Find the inverse of f (x) = 3x + 2.
1 Rewrite as ‘y =’y = 3x + 2
2 Switch x and yx = 3y + 2
3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2
3 = y
4 Rewrite as ‘f−1(x) =’f−1(x) = x−2
3
![Page 36: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/36.jpg)
Finding Inverses
This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.
Example
Find the inverse of f (x) = 3x + 2.
1 Rewrite as ‘y =’y = 3x + 2
2 Switch x and yx = 3y + 2
3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2
3 = y4 Rewrite as ‘f−1(x) =’
f−1(x) = x−23
![Page 37: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/37.jpg)
Plot of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)
Domain of f−1(x)? R
![Page 38: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/38.jpg)
Plot of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)
Domain of f−1(x)? R
![Page 39: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/39.jpg)
Plot of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)
Domain of f−1(x)? R
![Page 40: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/40.jpg)
Plot of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)
Domain of f−1(x)?
R
![Page 41: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/41.jpg)
Plot of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)
Domain of f−1(x)? R
![Page 42: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/42.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 43: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/43.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 44: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/44.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 45: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/45.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 46: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/46.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 47: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/47.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 48: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/48.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 49: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/49.jpg)
Finding Inverses
Example
Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.
How do we know this f (x) is not invertible without restricting thedomain?
It is not 1-1 ...
y = x2 − 3
x = y2 − 3
x + 3 = y2
y = ±√
x + 3
Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =
√x + 3 and D(f−1) = [−3,∞).
![Page 50: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/50.jpg)
Graph of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)
![Page 51: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/51.jpg)
Graph of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)
![Page 52: Section 1.6 Inverses - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/inverses.pdfInverses Definition If f(a) = b, then the function f 1 is the inverse function](https://reader033.vdocuments.us/reader033/viewer/2022050518/5fa1b601f9ab6e2c011f0f4d/html5/thumbnails/52.jpg)
Graph of f and f−1
-5 -4 -3 -2 -1 0 1 2 3 4 5
5
4
3
2
1
-5
-4
-3
-2
-1
f (x)
f−1(x)