### Functions

### Lesson 3: Inverse Function

(1) The meaning of "inverse?

Let f be a mapping from set X to set Y. The inverse of f, denoted by represents the "reverse" mapping from Y back to X.

Example 1

Let y = f (x), with f (x) = 2x + 1.

f maps x = 3 to y = 7.

Then "reverse" maps y = 7 back to x = 3.

Note: The inverse mapping can be obtained by expressing x

in terms of y, i.e.

(2) Existence of inverse function

If a function f is 1 to 1, then it will have an inverse function, i.e. f must satisfy the horizontal line test.

(3) Domain, Rule & Range

To find the rule of , let y = f (x) and rearrange x in terms of y. If there are multiple expressions for x, only select the one which satisfies .

Example 2

Let the function f be defined as follows:

Find .

Solution

(4) Graph of inverse functions

The graph of a function and its inverse are reflections of each other about the line y = x. Essentially, all the original x and y coordinates will be interchanged.

Example 3

Note: If the graph of a function f and its inverse intersect along the line y = x, we can obtain the point of intersection by solving f (x) = x.