Section 1.6: Inverse Functions

This section provides a detailed exploration of inverse functions and their properties in calculus. The content covers fundamental concepts of inverse functions, including their graphical representation and calculation methods.

Definition: An inverse function f⁻¹$x$ is a function that "undoes" the original function f$x$, where the graph of f⁻¹$x$ is the reflection of f$x$ about the line y=x.

Example: If f(3) = 11, then f⁻¹(11) = 3, demonstrating how inverse functions reverse the input-output relationship.

Highlight: To find the inverse of a function algebraically:

1. Replace f$x$ with y
2. Interchange x and y
3. Solve for y
4. Replace y with f⁻¹$x$

Vocabulary: One-to-one function refers to a function that passes the horizontal line test, meaning each y-value corresponds to exactly one x-value.

Example: For the function f$x$ = 7x - 12, its inverse is calculated by:

1. Writing y = 7x - 12
2. Switching x and y
3. Solving for y: x = 7y - 12 → y = $x + 12$/7
4. Therefore, f⁻¹$x$ = $x + 12$/7

Highlight: When finding how to find domain of inverse function, it's crucial to consider restrictions that ensure the original function is one-to-one.

Example: For f$x$ = $x + 1$², the domain must be restricted to (-∞, -1] to ensure the function is one-to-one, resulting in f⁻¹$x$ = -√x - 1.