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Section 1.6 Inverses
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?
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?
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?
Important Terms
Let f : X → Y be a function mapping X to Y .
DefinitionX is the domain of f .
Important Terms
Let f : X → Y be a function mapping X to Y .
DefinitionX is the domain of f .
Important Terms
What is the Y? (blue region here)
DefinitionY is the codomain of f .
Important Terms
What is the Y? (blue region here)
DefinitionY is the codomain of f .
Important Terms
So what is the yellow region here?
DefinitionThe range of f is the set of all f (x) where x ∈ X.
Important Terms
So what is the yellow region here?
DefinitionThe range of f is the set of all f (x) where x ∈ X.
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.
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.
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.
In Terms of the Elements
Definitionx is called the preimage of y = f (x).
Why We Bring This Up
A function can only have an inverse if it is 1-1. Do we rememberwhat 1-1 means?
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.
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.
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 ...
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 ...
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,∞).
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,∞).
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,∞).
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,∞).
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,∞).
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,∞).
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,∞).
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,∞).
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)
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)
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)