1. Section 1.8
INVERSE
FUNCTIONS

2. Function – is a relation that takes a value from set X and maps it into exactly one value from Set Y1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 10 5
Set X
Set Y
An inverse function would reverse that process, and map Set Y back into Set X

3. If we map what we get out of the function back,
we will not always have a function going back. 1 2 2 4 3 6 4 8 5
In this example, going back, value 6 maps into both 3 and 5, therefore the reverse relation is NOT a function. Functions of this type are called many-to-one functions.
Only functions that pair the y value (value in the range) with only one x will become functions going back the other way. These functions are called one-to-one functions.

4. This IS NOT a one-to-one function because y value is paired with more than one x value ( y is used more than once). 1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 5 10
This IS a one-to-one function. Each x is paired with only one y and each y is paired with only one x.
Only one-to-one functions will have inverse functions, meaning that mapping back to the original values will be also a function.

5. Function NOT a function Function
Recall: to determine whether a relation is a function (given a graph), we can use a vertical line test.
If a vertical line intersects the graph of an equation more than one time, the equation graphed is
NOT a function.
Function
NOT a function
Function

6. To be a one-to-one function, each y value could only be paired with one x. Let’s look at a couple of graphs.
Look at a y value, Ex. y = Is there only one corresponding x value?
For any y value, Ex. y = - 1, y = 1, etc., how many corresponding x values do you see?
This is a many-to-one function
This IS a one-to-one function

7. If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function.
This is NOT a one-to-one, not a function
This is NOT a one-to-one function
This is a one-to-one function

8. These functions are reflections of each other about the line y = x
Notice that the x and y values traded places for the function and its inverse.
These functions are reflections of each other about the line y = x
Let’s consider the function , compute some values, and graph them.
This means “inverse function”
x f (x)
(2,8)
(8,2)
x f -1(x)
Let’s take obtained values of the function and put them into the inverse function, then plot the points
(-8,-2)
(-2,-8)
Yes, so it will have an inverse function
Is this a one-to-one function?
What will “undo” a cube?
A cube root

9. Graphs of a function f and its inverse f -1
are reflections (symmetric) about the line y = x,
and the x and y values have traded places.
The domain of the function f is the range of the inverse f -1.
The range of the function f is the domain of the inverse f -1.
Algebraically, we can tell if two functions are inverses of each other if we substitute one into the other and that “undoes” the function.
DEFINITION
f and g are inverse functions if and only if their compositions for each x in the domain of g and for each x in the domain of f
and

10. Verify that the functions f and g are inverses of each other.
If we graph f(x) = (x – 2)2, it is a parabola shifted right 2.
Is this a one-to-one function?
This would not be one-to-one function unless we restrict the domain and
only consider the function where
x is greater than or equal to 2,
so we have a one-to-one function.

11. are inverses of each other.
Verify that the functions f and g are inverses of each other algebraically:
Since both compositions are equal to x, according to the definition of inverse functions, f and g
are inverses of each other.

12. confirm inverses by finding compositions
Steps for Finding the Inverse of a One-to-One Function (the Algorithm)
Use correct notation
y = f -1(x)
Solve for y
And,
confirm inverses by finding compositions
Trade x and y places
Replace f(x) with y

13. *Ensure f(x) is one-to-one first. Domain may need to be restricted.
Confirm inverse by finding
Find the inverse of
Use correct notation
y = f -1(x) or
Solve for y
Trade x and y places
Replace f(x) with y
*Ensure f(x) is one-to-one first. Domain may need to be restricted.