Overview
This lecture explains inverse functions, focusing on their definitions, conditions for existence, and methods for finding the inverse both with ordered pairs and algebraic equations.
Definitions and Properties of Inverse Functions
- The inverse of a function undoes the operation of the original function.
- The domain of the inverse is the range of the original function, and vice versa.
- The inverse of function f is denoted as (f^{-1}).
- A function must be one-to-one (bijective) to have an inverse.
Finding the Inverse of a Function
- To find the inverse: Write the function as (y = f(x)), swap x and y, then solve for y.
- Example: For (f(x) = 3x + 1), the inverse is (f^{-1}(x) = (x - 1)/3).
- Example: For (g(x) = x^3 - 2), the inverse is (g^{-1}(x) = \sqrt[3]{x + 2}).
- Example: For (f(x) = \frac{2x + 1}{3x - 4}), after rearranging and solving, the inverse is (f^{-1}(x) = \frac{4x + 1}{3x - 2}).
- Swapping x and y in ordered pairs also gets the inverse set.
One-to-One Functions and Non-Invertibility
- Only one-to-one functions pass the horizontal line test and have inverses.
- Quadratic functions (e.g., (f(x) = x^2 + 4x - 2)) and absolute value functions (e.g., (f(x) = |3x|)) do not have inverses because they are not one-to-one.
Application Example: Temperature Conversion
- The function for converting Fahrenheit to Kelvin: (K(T) = \frac{5}{9}(T - 32) + 273.15).
- To find the inverse (Kelvin to Fahrenheit): (T(K) = \frac{9}{5}(K - 273.15) + 32).
Verifying Inverse Functions
- To check if two functions are inverses, use composition: (f(g(x)) = x) and (g(f(x)) = x).
- If both compositions result in x, the functions are inverses.
Key Terms & Definitions
- Inverse Function — A function that reverses the effect of the original function.
- One-to-One Function — A function where each output corresponds to exactly one input.
- Horizontal Line Test — A test to check if a function is one-to-one.
Action Items / Next Steps
- Practice finding inverses for various functions.
- Identify if given functions are one-to-one before attempting to find inverses.
- Complete assigned problems on verifying inverses using composition.