The Main Idea

• Definition of Inverse Functions:
  • An inverse function [latex]f^{-1}(x)[/latex] “undoes” what the original function [latex]f(x)[/latex] does
  • Input and output are swapped: if [latex]f(a) = b[/latex], then [latex]f^{-1}(b) = a[/latex]
• Notation:
  • [latex]f^{-1}(x)[/latex] represents the inverse of [latex]f(x)[/latex]
  • The “-1” is not an exponent; [latex]f^{-1}(x) \neq \frac{1}{f(x)}[/latex]
• Graphical Representation:
  • The graph of [latex]f^{-1}(x)[/latex] is a reflection of [latex]f(x)[/latex] over the line [latex]y = x[/latex]
  • If [latex](a, b)[/latex] is on [latex]f(x)[/latex], then [latex](b, a)[/latex] is on [latex]f^{-1}(x)[/latex]
• Conditions for Inverse Functions:
  • The original function must be one-to-one (injective)
  • Each output corresponds to exactly one input
  • Passes the horizontal line test
• Verifying Inverse Functions:
  • [latex]f(f^{-1}(x)) = x[/latex] and [latex]f^{-1}(f(x)) = x[/latex]
  • These identities should hold for all [latex]x[/latex] in the domain

The Main Idea

• Relationship between function and inverse domains/ranges:
  • Range of [latex]f(x)[/latex] = Domain of [latex]f^{-1}(x)[/latex]
  • Domain of [latex]f(x)[/latex] = Range of [latex]f^{-1}(x)[/latex]
• One-to-one functions:
  • Only one-to-one functions have inverse functions
  • Pass the horizontal line test
  • Identifying one-to-one functions:
    • Graphical method: Horizontal line test
    • Algebraic method: [latex]f(a) = f(b)[/latex] implies [latex]a = b[/latex]
• Restricting domains:
  • Non-one-to-one functions can be made invertible by restricting their domains
  • The restricted domain becomes the range of the inverse function
  • Domain restriction:
    • Choose a portion of the function that is one-to-one
    • Ensures the inverse is a valid function
• Uniqueness of inverse functions:
  • A function has only one inverse on a given domain

How to Find Domain and Range of Inverse Functions

1. For one-to-one functions:
   • Write the range of [latex]f(x)[/latex] as the domain of [latex]f^{-1}(x)[/latex]
   • Write the domain of [latex]f(x)[/latex] as the range of [latex]f^{-1}(x)[/latex]
2. For restricted functions:
   • Identify the restriction that makes the function one-to-one
   • The restricted domain of [latex]f(x)[/latex] becomes the range of [latex]f^{-1}(x)[/latex]
   • The range of the restricted [latex]f(x)[/latex] becomes the domain of [latex]f^{-1}(x)[/latex]

The Main Idea

• Inverting tabular functions:
  • Interchange domain and range
    • Domain of [latex]f(x)[/latex] = Range of [latex]f^(-1)(x)[/latex]
    • Range of [latex]f(x)[/latex] = Domain of [latex]f^(-1)(x)[/latex]
  • Swap inputs and outputs
• Evaluating inverses from graphs:
  • Use vertical axis for inverse input
    • Vertical extent of [latex]f(x)[/latex] = Domain of [latex]f^(-1)(x)[/latex]
  • Use horizontal axis for inverse output
    • • Horizontal extent of [latex]f(x)[/latex] = Range of [latex]f^(-1)(x)[/latex]
• Finding inverse functions from formulas:
  • Solve for [latex]x[/latex] in terms of [latex]y[/latex]
  • Replace [latex]y[/latex] with [latex]f^(-1)(x)[/latex]

Methods for Finding and Evaluating Inverse Functions

1. For tabular functions:
   • Swap the input and output columns
   • Read the new table for the inverse function values
2. For graphical functions:
   • Use the [latex]y[/latex]-axis of the original function as the input for the inverse
   • Read the corresponding [latex]x[/latex]-value as the output of the inverse
3. For formula-based functions:
   1. Verify the function is one-to-one
   2. Replace [latex]f(x)[/latex] with [latex]y[/latex]
   3. Interchange [latex]x[/latex] and [latex]y[/latex]
   4. Solve for [latex]y[/latex]
   5. Rename the resulting expression as [latex]f^(-1)(x)[/latex]

The Main Idea

• Graphical relationship between a function and its inverse:
  • Reflection over the line [latex]y = x[/latex] (identity line)
• Domain and range relationships:
  • Domain of [latex]f(x)[/latex] becomes range of [latex]f^{-1}(x)[/latex]
  • Range of [latex]f(x)[/latex] becomes domain of [latex]f^{-1}(x)[/latex]
• Restricting domains for invertibility:
  • Some functions need domain restrictions to be one-to-one
• Functions that are their own inverses:
  • Special cases where [latex]f(x) = f^{-1}(x)[/latex]

Key Concepts

• Reflection principle:
  • Graph of [latex]f^{-1}(x)[/latex] is the reflection of [latex]f(x)[/latex] over [latex]y = x[/latex]
• One-to-one functions:
  • Necessary for a function to have an inverse
  • Pass the horizontal line test
• Identity line:
  • The line [latex]y = x[/latex]
  • Acts as the “mirror” for reflecting graphs
• Restricted domains:
  • Make non-one-to-one functions invertible
  • Example: [latex]f(x) = x^2[/latex] restricted to [latex][0, \infty)[/latex]

Techniques for Graphing Inverse Functions

1. Identify key points on the original function:
   • [latex]y[/latex]-intercepts become [latex]x[/latex]-intercepts for the inverse
   • [latex]x[/latex]-intercepts become [latex]y[/latex]-intercepts for the inverse
2. Reflect these key points over [latex]y = x[/latex]
3. Consider domain and range:
   • Vertical asymptotes become horizontal asymptotes
   • Horizontal asymptotes become vertical asymptotes
4. Sketch the inverse function:
   • Connect the reflected points
   • Ensure the graph passes the vertical line test
5. Verify inverse relationship:
   • Check if composition yields the identity function