Unit 2: Inverse Functions

An inverse function reverses the action of the original function. Understanding inverse functions is essential for solving equations and understanding symmetry in graphs.

Definition

An inverse function reverses the action of the original function. If \(f(x)\) is the original function, the inverse \(f^{-1}(x)\) satisfies:

\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x. \]

Video Lessons

Understanding Inverse Functions
More Videos
Introduction to Inverse Functions Finding Inverses of Linear Functions Inverses of Quadratic Functions Graphing Inverse Functions

Key Concepts

  1. Domain and Range:
    • The domain of \(f(x)\) becomes the range of \(f^{-1}(x)\).
    • The range of \(f(x)\) becomes the domain of \(f^{-1}(x)\).
  2. Reflection Property:
    • The graph of \(f(x)\) and \(f^{-1}(x)\) are symmetrical about the line \(y = x\).
  3. One-to-One Functions:
    • Only one-to-one functions have inverses that are also functions.
    • A function is one-to-one if it passes the horizontal line test.

Steps to Find the Inverse

  1. Replace \(f(x)\) with \(y\).
  2. Swap \(x\) and \(y\).
  3. Solve for \(y\).
  4. Replace \(y\) with \(f^{-1}(x)\).

Worked Examples

Example 1: Linear Function

Find the inverse of \(f(x) = 2x + 6\).

  1. Replace \(f(x)\) with \(y\): \(y = 2x + 6\).
  2. Swap \(x\) and \(y\): \(x = 2y + 6\).
  3. Solve for \(y\): \[ x - 6 = 2y \implies y = \frac{x - 6}{2}. \]
  4. Write as \(f^{-1}(x)\): \[ f^{-1}(x) = \frac{x - 6}{2}. \]
Example 2: Quadratic Function

Find the inverse of \(f(x) = 2x^2\) with \(x \geq 0\).

  1. Replace \(f(x)\) with \(y\): \(y = 2x^2\).
  2. Swap \(x\) and \(y\): \(x = 2y^2\).
  3. Solve for \(y\): \[ y^2 = \frac{x}{2} \implies y = \sqrt{\frac{x}{2}}. \]
  4. Write as \(f^{-1}(x)\): \[ f^{-1}(x) = \sqrt{\frac{x}{2}}, \quad x \geq 0. \]

Graphical Representation

  1. Plot \(f(x)\) using its key points.
  2. Reflect the graph across the line \(y = x\) to find \(f^{-1}(x)\).
  3. Check if \(f^{-1}(x)\) is a function by using the vertical line test.

Important Notes

  • The notation \(f^{-1}(x)\) does not mean \(\frac{1}{f(x)}\).
  • Some inverses (e.g., of quadratic functions) are not functions unless their domains are restricted.

Would you like further practice problems or assistance with specific types of functions?

Examples and Practice Questions

  1. Find the inverse of \(f(x) = 3x - 5\).
  2. Determine the inverse of \(f(x) = \frac{1}{x}\).
  3. Find the inverse of \(f(x) = x^3\).
  4. Given \(f(x) = 4x + 7\), find \(f^{-1}(x)\).
  5. Find the inverse of \(f(x) = \sqrt{x}\), \(x \geq 0\).

Flashcards

What is an inverse function?

A function that reverses the action of the original function.

How do you find an inverse function?

Swap \(x\) and \(y\) in the equation and solve for \(y\).

What is the reflection property?

The graphs of \(f(x)\) and \(f^{-1}(x)\) are symmetrical about the line \(y = x\).

Practice Questions

  1. Find the inverse of \(f(x) = 5x - 10\).
  2. Determine if the function \(f(x) = x^2\) has an inverse function (without restricting domain).
  3. Find the inverse of \(f(x) = \frac{2}{x}\).
  4. Given \(f(x) = \ln(x)\), find \(f^{-1}(x)\).

Quick Quiz

1. Find the inverse of \(f(x) = 3x + 4\).

2. Determine the inverse of \(f(x) = \sqrt{x}\), \(x \geq 0\).

Horizontal Line Test Demo

Use the slider to move a horizontal line over the graph of \(f(x) = x^2\). If any horizontal line intersects the graph more than once, the function is not one-to-one and does not have an inverse function without restriction.

Check Your Understanding: Inverse Calculation

Given \(f(x) = 3x - 5\), enter what you think is \(f^{-1}(x)\):

Function-Inverse Matching Game

Drag the inverse function onto the correct original function:

Functions
f(x)=x+2
f(x)=2x
f(x)=x-1
Inverse Slots
Drop inverse of x+2 here
Drop inverse of 2x here
Drop inverse of x-1 here
Inverses
f⁻¹(x)=x-2
f⁻¹(x)=x/2
f⁻¹(x)=x+1

Summary

  • An inverse function reverses the action of the original function.
  • Only one-to-one functions have inverses that are also functions.
  • The domain and range of a function swap when finding its inverse.
  • The graphs of a function and its inverse are symmetrical about the line \(y = x\).
  • Understanding inverse functions is crucial for solving equations and analyzing function behavior.
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