# Operations with Functions

### Other Methods of Finding Inverses

#### Finding the Inverse of a Function by Isolating f -1(x)

There is another method we can use to find the inverse of a function. Taking the inverse "reverses" x and f (x) . Thus, in the original function, substitute " x " for " f (x) " and substitute " f -1(x) " for " x ". Then, solve for f -1(x) using inverse operations in the usual manner.

Example 1: f (x) = .

1. Substitute: x =
2. Solve for f -1(x) :

 5x = 2(f -1(x)) - 1 5x + 1 = 2(f -1(x)) = f -1(x) f -1(x) =

3. Check: f -1(f (x)) = f -1() = = = = x

Example 2: f (x) = , x≠1 , f (x) > 0 .

1. Substitute: x =
2. Solve for f -1(x) :

 x(f -1(x) - 1)2 = 1 (f -1(x) - 1)2 = f -1(x) - 1 = f -1(x) = + 1

3. Check: f -1(f (x)) = f -1( = +1 = + 1 = (x - 1) + 1 = x .

Domain of f -1 : x > 0 . Range of f -1 : f -1(x)≠1 .

#### Finding the Inverse of a Function by Graphing

We can also find the inverse of a function by graphing. The inverse of a function is a reflection of that function over the line y = x . In other words, all points (x, y) = (a, b) become (x, y) = (b, a) . The x and y coordinates of each point switch:

Inverse of a Graph

To find the inverse of a function, reflect the function over the line y = x . Or, find several points on the graph of y = f (x) , switch their x and y coordinates, and graph the resulting points. Connect these points with a line or curve that mirrors the line or curve of the original function.

Example: Find the inverse y = 2x - 1 by graphing:

Inverse of y = 2x - 1

## Take a Study Break

### What's your Pretty Little Liars name?

Take this quiz to find out!

### Which young actress just got married?

Click to find out!

### Cat bearding WINS THE INTERNET

Have you seen this yet?

### Scary movies with funny posters

These. Are. Hilarious.

### Geeky Actors: Then and Now

Travel back in time!

### Villains We Want These Actresses to Play

From super cute to super bad!

### 10 Movies Better Than Their Books

What do you think?