Inverse Functions

Introduction to Inverse Functions

• Review: Properties and behaviors of inverse functions.
• Objective: Learn to take derivatives of inverse functions.
• Key Condition: A function must be one-to-one to have an inverse.
  • One-to-one: Every input has a unique output.
  • Horizontal Line Test: Used to determine if a function is one-to-one.

Understanding Inverse Functions

• Inverse Definition: Switch inputs (x) and outputs (y).
• Importance: A non-one-to-one function loses its "function" property when inverted.
• Notation: "f inverse" of x is denoted as f^-1(x), not an exponent.

Checking Inverses

• Composition Test: Functions f and f inverse undo each other.
  • Composition of f(f^-1(x)) or f^-1(f(x)) should return x.

Finding Inverses

• Steps:
  1. Confirm function is one-to-one.
  2. Express the function as y = f(x).
  3. Swap x and y, then solve for y.
  4. Verify inverse using composition.

Graphical Relationships

• Graphs: A function and its inverse are reflections over the line y = x.
• Continuity: If a function is continuous, its inverse will also be continuous.
  • Domain of f becomes the range of f inverse.
  • Range of f becomes the domain of f inverse.

Differentiability

• Conditions: If a function f is differentiable, f inverse is also differentiable.

Evaluating Inverses

• Method:
  1. Set f(x) = a and solve for x to find f inverse(a).
  2. Use x found as the result of f inverse(a).

Derivatives of Inverse Functions

• Finding Derivatives without Explicit Inverses:
  • Use the formula for derivative of an inverse, G' = 1/(f'(G(x))).

Example Problems

• Problem 1: Verify a point is on the inverse by plugging into the function.
• Problem 2: Find derivatives at specific points using the derivative formula.

Summary

• Understanding the properties and calculations related to inverse functions is crucial for calculus, especially for functions where directly finding inverses is complex.
• Next Topic: Explore exponential functions, which are inverses of logarithmic functions.