1. 4-5 Inverse Functions

2. First, lets define something.
A One to One Function – is a function where each y has a unique x.
(1, 1) (2, 2), (3, 5), (4,12) is one-to-one
(1, 1) (2, 2), (3, 2), (4,12) is not.

3. What was the graphic way to test functionality?
For regular functions, you did a vertical line test.
For one to one functions, they ALSO pass a horizontal line test. That is, as long as each x has its own unique y that no other x has, it is one to one.
The connection will be made later.

4. What is an inverse function?
Two functions are said to be inverses iff
f(g(x)) = g(f(x)) = x. That is, the composite of a function with its own inverse is x.
Now what does that mean? It means that they are functions that “negate” each other, such as and

5. So, lets experiment Let and What is f(g(x))? How about g(f(x))?
Find f(0), f(1), f(2) and f(3).
Now find g(0), g(1), g(4) and g(9).
Make a table for both functions.
Do you see it yet??

6. So what is going on? That is an inverse – for each ordered pair
(x,y) the inverse creates the ordered pair (y,x).
Notation for an inverse
if it is a function
if it is NOT a function

7. The visual of an inverse
Graph the two data sets on a single coordinate axis. What do you see?
If functions are inverses, then they are reflections across the line y = x.

8. f(x)
(1,2)
f-1(x)
(2,1)

9. How do I FIND an inverse?
The easiest way is this: Replace every x with a y and every y with an x, then solve for y.
How to test? Take the composite with its possible inverse – if you get x, then its correct.

10. Examples