Inverse Functions: Overview

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Inverse Functions: Section Overview

What You'll Learn in §6.1

This section covers four core skills. Master them in order for best results.

Learning Path

graph TD
    A["1. One-to-One<br/>Functions"]
    B["2. Inverse Function<br/>Definition"]
    C["3. Finding Inverse<br/>Functions"]
    D["4. Graphing<br/>Inverses"]
    E["5. Derivative of<br/>Inverse Functions"]

    A --> B
    B --> C
    B --> D
    C --> E
    D --> E

    style A fill:#e0f2fe
    style B fill:#e0f2fe
    style C fill:#fef3c7
    style D fill:#fef3c7
    style E fill:#fce7f3

    click A "one-to-one-functions.html"
    click B "inverse-function-definition.html"
    click C "finding-inverse.html"
    click D "graphing-inverse-functions.html"
    click E "derivative-of-inverse-functions.html"

Color key: Blue = Foundational | Yellow = Intermediate | Pink = Advanced

Skill Pages

Foundational Skills (Start Here)

Skill	Time	What You'll Learn
One-to-One Functions	15 min	When does a function have an inverse? The horizontal line test.
Inverse Function Definition	15 min	What $f^{-1}$ means, domain/range swap, cancellation equations.

Core Skills

Skill	Time	What You'll Learn
Finding Inverse Functions	20 min	The 3-step algebraic method to find $f^{-1}$.
Graphing Inverse Functions	15 min	Reflecting graphs across $y = x$.

Calculus Application

Skill	Time	What You'll Learn
Derivative of Inverse Functions	20 min	The formula $(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$.

Quick Self-Assessment

Before starting this section, you should be able to:

• [ ] Define what a function is (input → unique output)
• [ ] Evaluate functions from formulas, tables, and graphs
• [ ] Compose functions: compute $f(g(x))$
• [ ] Differentiate polynomial and trigonometric functions

The Big Picture

Inverse functions "undo" what original functions do. This concept is the foundation for:

• Logarithms (inverse of exponentials)
• Inverse trigonometric functions ($\arcsin$, $\arccos$, $\arctan$)
• Solving equations where the variable is "trapped" inside a function

The derivative formula for inverses will be used repeatedly in §6.2 through §6.6 to derive the derivatives of $\ln x$, $\arcsin x$, and other inverse functions.

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Chapter 5 Review	Chapter 6	Exponential Functions (§6.2)

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Last updated: 2026-01-23