# Operations with Functions

### Inverse Functions

#### Inverse Functions

Two functions f and g are inverse functions if f o g(x) = x and g o f (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 f o g(x) = f (g(x)) = f ( x) = 2( x) = x and g o f (x) = g(f (x)) = g(2x) = (2x) = x . Similarly, f (x) = x + 1 and g(x) = x - 1 are inverse funcions because f o g(x) = f (g(x)) = f (x - 1) = (x - 1) + 1 = x and g o f (x) = g(f (x)) = g(x + 1) = (x + 1) - 1 = x . h(x) = 3x - 1 and j(x) = are inverse functions because h o j(x) = h(j(x)) = h() = 3() - 1 = x + 1 - 1 = x and j o h(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 .

Check:
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 .

Example 2: Find the inverse of f (x) = , x≥2 (we must restrict the domain because f (x) is undefined for x < 2 ).

Original function:

1. Subract 2.
2. Take the square root.
New function:
1. Square.
Thus, f -1(x) = x 2 + 2 .

Check:
f -1(f (x)) = f -1() = ()2 + 2 = (x - 2) + 2 = x .
f (f -1(x)) = f (x 2 +2) = = = x .

When we take the inverse of a function, the domain and range switch. In example 2, the domain of f is x≥2 and the range of f is f (x)≥ 0 . Thus, the domain of f -1 is x≥ 0 and the range of f -1 is f -1(x)≥2 .

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