### Inverse Functions

Two functions f and g are inverse functions if fog(x) = x and gof (x) = x for all values of x in the domain of f and g.

For instance, f (x) = 2x and g(x) = x are inverse functions because fog(x) = f (g(x)) = f ( x) = 2( x) = x and gof (x) = g(f (x)) = g(2x) = (2x) = x. Similarly, f (x) = x + 1 and g(x) = x - 1 are inverse funcions because fog(x) = f (g(x)) = f (x - 1) = (x - 1) + 1 = x and gof (x) = g(f (x)) = g(x + 1) = (x + 1) - 1 = x. h(x) = 3x - 1 and j(x) = are inverse functions because hoj(x) = h(j(x)) = h( ) = 3( ) - 1 = x + 1 - 1 = x and joh(x) = j(h(x)) = j(3x - 1) = = = x.

The inverse of a function f (x) is denoted f-1(x).

### Finding the Inverse of a Function by Reversing Operations

The trick to finding the inverse of a function f (x) is to "undo" all the operations on x in reverse order.

The function f (x) = 2x - 4 has two steps:

1. Multiply by 2.
2. Subtract 4.
Thus, f-1(x) must have two steps:
2. Divide by 2.
Consequently, f-1(x) = .
We can verify that this is the inverse of f (x):
f-1(f (x)) = f-1(2x - 4) = = = x.

f (f-1(x)) = f ( ) = 2( ) - 4 = (x + 4) - 4 = x.

Example 1: Find the inverse of f (x) = 3(x - 5).

Original function:

1. Subtract 5.
2. Multiply by 3.
New function:
1. Divide by 3.
Thus, f-1(x) = + 5.
f-1(f (x)) = f-1(3(x - 5)) = + 5 = (x - 5) + 5 = x.
f (f-1(x)) = f ( +5) = 3(( +5) - 5) = 3( ) = x.