One-to-One and Inverse Functions

One-to-One Functions

• A function is one-to-one if any two different inputs correspond to different outputs.
• Definition: If f(x₁) = f(x₂) implies x₁ = x₂, then f is one-to-one.

Horizontal-Line Test

• If any horizontal line intersects the graph at most once, the function is one-to-one.

Increasing/Decreasing Functions

• A function that is strictly increasing or strictly decreasing on an interval is one-to-one on that interval.

Inverse Functions

• For a one-to-one function f, there exists an inverse function, denoted f⁻¹.
• To find the inverse:
  1. Interchange x and y in the equation.
  2. Solve for y to get f⁻¹.

Procedure for Finding the Inverse

1. Interchange x and y in f(x).
2. Solve for y to find f⁻¹.
3. Verify by checking f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Graphs of Functions and Their Inverses

• The graph of f and f⁻¹ are symmetric relative to the line y = x.

Summary

1. A one-to-one function has an inverse.
2. The domain of f is the range of f⁻¹ and vice versa.
3. Verify f⁻¹ is the inverse by the stated conditions.
4. The graphs of f and f⁻¹ show symmetry around y = x.