1. Section 4.1
Inverse Functions

2. CONCEPT OF AN INVERSE FUNCTION
Idea: An inverse function takes the output of the “original” function and tells from what input it resulted.
Note that this really says that the roles of x and y are reversed.

3. MATHEMATICAL DEFINITION OF INVERSE FUNCTIONS
In the language of function notation, two functions f and g are inverses of each other if and only if

4. NOTATION FOR THE INVERSE FUNCTION
We use the notation
for the inverse of f(x).
NOTE:
does NOT mean

5. ONE-TO-ONE FUNCTIONS
A function is one-to-one if for each y-value there is only one x‑value that can be paired with it; that is, each output comes from only one input.

6. ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS
Theorem: A function has an inverse if and only if it is one-to-one.

7. TESTING FOR A ONE-TO-ONE FUNCTION
Horizontal Line Test: A function is one-to-one (and has an inverse) if and only if no horizontal line touches its graph more than once.

8. GRAPHING AN INVERSE FUNCTION
Given the graph of a one-to-one function, the graph of its inverse is obtained by switching x- and y-coordinates.
The resulting graph is reflected about the line y = x.

9. FINDING A FORMULA FOR AN INVERSE FUNCTION
To find a formula for the inverse given an equation for a one-to-one function:
Replace f (x) with y.
2. Interchange x and y.
3. Solve for y
4. Replace y with f -1(x).
Solve the resulting

10. Examples

11. Answers