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:
- Replace f(x) with y
- Interchange x and y
- Solve for y
- 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:
- Writing y = 7x - 12
- Switching x and y
- Solving for y: x = 7y - 12 → y = (x + 12)/7
- 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.
