Inverse Functions

Definition and Properties

The inverse of a function is a function that reverses the effect of the original function. If f is a function, its inverse, denoted f-1, satisfies the property that applying f and then f-1 returns the original input:

• Key Property: and for all in the domain of f-1 and f respectively.

• Existence: Not all functions have inverses. A function must be one-to-one (bijective) to have an inverse.

Example: If and , then and are inverses of each other because:

Finding the Inverse of a Function

To find the inverse of a function f, follow these steps:

1. Replace with in the equation for $f(x)$.

2. Interchange and .

3. Solve for . If the resulting equation defines $y$ as a function of , then has an inverse function.

Example: Find the inverse of .

• Step 1:

• Step 2: Interchange and :

• Step 3: Solve for :

• Thus,

Horizontal Line Test for Inverse Functions

The Horizontal Line Test is used to determine if a function has an inverse. If every horizontal line intersects the graph of a function at most once, then the function is one-to-one and has an inverse.

• Application: If a function fails the horizontal line test, it does not have an inverse function.

Graphs of Functions and Their Inverses

The graph of a function and its inverse are reflections of each other across the line . To draw the graph of the inverse function, reflect each point of the original function to .

• Example: If the graph of passes through , the graph of passes through .

Application: Use the provided coordinate grid to plot both and , reflecting points across the line .

Summary Table: Steps to Find the Inverse

Additional info: The notes briefly mention graphical examples and the horizontal line test, which are fundamental for understanding inverse functions in College Algebra.