1.5 Inverse Functions

Definition of the Inverse of a Function

Let f and g be two functions such that
f(g(x))=x for every x in the domain of g

and

g(f(x))=x For every x in the domain of f

Under these conditions, the function g is the inverse of the function f. The function g is denoted by

, and the range of f must be equal to the domain of

.

Expressing an inverse function

An inverse function is expressed by the term

, It is not referring to the reciprocal of f(x), which is

Finding the Inverse of Functions Informally
Ex. Find the inverse of f(x)=4x. Then verify that both

and

are equal to the identity function.

Solution
The given function multiplies each input by 4 "undo" this function. you need to divide each input by 4. So, the inverse function of f(x)=4x is

Finding the Inverse of a Function
1. Use the Horizontal line test to decide whether f has an inverse.
2. In the equation for f(x), replace f(x) by y.
3. Interchange the roles of x and y, and solve for y
4. Replace y by

(x) in the new equation.
5. verify that f and

are inverses of each other by showing that

= x and