Inverse Functions

Definition and Properties

An inverse function reverses the effect of the original function. If is a function, its inverse satisfies and for all in the domain of and respectively.

• One-to-One Function: A function is one-to-one (injective) if each output is produced by exactly one input. This is necessary for a function to have an inverse.

• Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once.

• Invertibility: Only one-to-one functions are invertible. If a function fails the horizontal line test, it does not have an inverse.

Example: The function is one-to-one and invertible. The function (on ) is not one-to-one and not invertible.

Justifying Invertibility

• If a function passes the horizontal line test, it is invertible.

• If a function fails the horizontal line test, it is not invertible.

Example: Given a graph, if every horizontal line crosses the graph at most once, the function is invertible.

Finding the Inverse of a Function

Step-by-Step Process

1. Replace with .

2. Interchange and .

3. Solve for .

4. Replace with .

Example 1: Find the inverse of .

• Step 1:

• Step 2:

• Step 3:

• Step 4:

Example 2: Find the inverse of .

• Step 1:

• Step 2:

• Step 3:

• Step 4:

Verifying Inverse Functions

Algebraic Verification

To verify that and are inverses, check:

• for all in the domain of

• for all in the domain of

Example: Let , .

Therefore, and are inverse functions.

Domain and Range of Inverse Functions

Relationship Between Domain and Range

• The domain of becomes the range of .

• The range of becomes the domain of .

Example: If has domain and range , then has domain and range .

Tables and Mapping Diagrams

Using Tables to Find Inverse Values

Given a table of values for , the inverse maps outputs back to their corresponding inputs.

• Example: because .

• Example: To solve , find such that .

Graphing Inverse Functions

Graphical Relationship

• The graph of is the reflection of the graph of across the line .

• To sketch , swap the and coordinates of each point on .

Example: If passes through , then passes through .

Summary Table: Properties of Invertible Functions

Worked Examples

Example 1: Find the Inverse

• Given , find .

• Step 1:

• Step 2:

• Step 3:

• Step 4:

Example 2: Verify Inverse Functions

• Let , .

Key Takeaways

• Only one-to-one functions are invertible.

• The inverse function undoes the action of the original function.

• To find the inverse, swap and and solve for .

• The graph of the inverse is a reflection over .

• Domain and range swap between a function and its inverse.