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⁻¹x is a function that "undoes" the original function fx, where the graph of f⁻¹x is the reflection of fx about the line y=x.
Example: If f3 = 11, then f⁻¹11 = 3, demonstrating how inverse functions reverse the input-output relationship.
Highlight: To find the inverse of a function algebraically:
- Replace fx with y
- Interchange x and y
- Solve for y
- Replace y with f⁻¹x
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 fx = 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⁻¹x = 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 fx = x+1², the domain must be restricted to −∞,−1]toensurethefunctionisone−to−one,resultinginf−1(x = -√x - 1.
