Inverse Functions: Finding the Equation of the Inverse

Functions

Inverse Functions

An inverse function reverses the effect of the original function. If a function $f$ maps $x$ to $y$, then its inverse $f^{-1}$ 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 $f(x)$ is a one-to-one function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Steps to Find the Inverse:

  1. Replace $f(x)$ with $y$.

  2. Solve the equation for $x$ in terms of $y$.

  3. Interchange $x$ and $y$ to write the inverse function $f^{-1}(x)$.

Example: Find the Inverse

Given the function $g(x) = x^3 + 1$, find the equation for the inverse function.

• Step 1: Replace $g(x)$ with $y$: $y = x^3 + 1$

• Step 2: Solve for $x$: $y - 1 = x^3$ $x = \sqrt[3]{y - 1}$

• Step 3: Interchange $x$ and $y$: $g^{-1}(x) = \sqrt[3]{x - 1}$

Final Answer:

• $g^{-1}(x) = \sqrt[3]{x - 1}$

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 $y = x$.