Functions

Inverse Functions

An inverse function reverses the effect of the original function. If a function maps to , then its inverse maps $y$ back to $x$. The process of finding the inverse involves solving for $x$ in terms of $y$ and then rewriting the result as a function of $x$.

• Definition: If is a one-to-one function, its inverse satisfies and .

• Steps to Find the Inverse:

  1. Replace with .

  2. Solve the equation for in terms of .

  3. Interchange and to write the inverse function .

Example: Find the Inverse

Given the function , find the equation for the inverse function.

• Step 1: Replace with :

• Step 2: Solve for :

• Step 3: Interchange and :

Final Answer:

Key Properties of Inverse Functions:

• Not all functions have inverses. A function must be one-to-one (pass the horizontal line test).

• The graph of an inverse function is a reflection of the original function across the line .