Inverse Functions Rules

Why This Matters

Inverse functions are one of those topics that keeps showing up throughout Algebra 2 and beyond—you'll need them for solving exponential and logarithmic equations, working with trigonometric functions, and understanding function composition. When you're tested on inverses, you're really being tested on your understanding of function behavior, domain and range relationships, and graphical transformations. The concept of "undoing" a function connects directly to solving equations, which is the heart of algebra.

Don't just memorize the steps for finding an inverse. You need to understand why we swap variables, how the graphs relate, and when a function even has an inverse in the first place. Every rule here ties back to one core idea: inverse functions reverse the input-output relationship. Master that concept, and the rest falls into place.

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The Core Concept: Reversing Input and Output

An inverse function takes each output of the original function and maps it back to its corresponding input—essentially running the function backward.

What an Inverse Function Does

• Inverses "undo" the original function—if $f(x)$ transforms $x$ into $y$, then $f^{-1}(y)$ transforms $y$ back into $x$
• The notation $f^{-1}(x)$ means "the inverse function of $f$," not $\frac{1}{f(x)}$—this is a common exam trap
• Example: if $f(x) = 2x$, then $f^{-1}(x) = \frac{x}{2}$ because dividing by 2 reverses multiplying by 2

Domain and Range Swap

• The domain of $f(x)$ becomes the range of $f^{-1}(x)$—input values of the original become output values of the inverse
• The range of $f(x)$ becomes the domain of $f^{-1}(x)$—this swap is automatic when you reverse the function
• Exam application: if you're given domain restrictions on $f(x)$, those become range restrictions on $f^{-1}(x)$

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Finding Inverse Functions Algebraically

The algebraic process for finding an inverse mirrors the conceptual idea: swap the roles of input and output, then isolate the new output variable.

The Three-Step Method

• Step 1: Replace $f(x)$ with $y$—this makes the variable swap clearer and easier to manipulate
• Step 2: Swap $x$ and $y$—this reverses the input-output relationship, which is the whole point of an inverse
• Step 3: Solve for $y$—your solution is $f^{-1}(x)$, expressed as a function of the new input

Verifying with Composition

• The composition test: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ must both be true
• This creates the identity function—applying a function and its inverse in either order returns the original input unchanged
• Use this to check your work: if your answer doesn't satisfy both compositions, you've made an error somewhere

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Graphical Relationships

The geometric relationship between a function and its inverse provides a powerful visual tool for understanding and checking your work.

Reflection Over $y = x$

• Graphs of $f(x)$ and $f^{-1}(x)$ are mirror images across the line $y = x$—this reflection property is testable and useful for sketching
• Coordinate swap: the point $(a, b)$ on $f(x)$ becomes $(b, a)$ on $f^{-1}(x)$
• Visual check: if your inverse graph doesn't reflect properly over $y = x$, something went wrong algebraically

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When Inverses Exist: The One-to-One Requirement

Not every function has an inverse that's also a function—the original must pass a specific test.

The One-to-One (Injective) Property

• A function is one-to-one when each output comes from exactly one input—no $y$-value is repeated for different $x$-values
• The Horizontal Line Test: if any horizontal line crosses the graph more than once, the function is not one-to-one and has no inverse function
• Why it matters: without one-to-one, the "reverse" would give multiple outputs for a single input, violating the definition of a function

Restricting Domains to Create Inverses

• Non-one-to-one functions can gain inverses by limiting their domain—you cut out the "repeat" portions
• Classic example: $f(x) = x^2$ fails the horizontal line test, but $f(x) = x^2$ with $x \geq 0$ is one-to-one with inverse $f^{-1}(x) = \sqrt{x}$
• Exam alert: always check whether domain restrictions are given—they change everything about the inverse

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Special Cases: Inverse Trigonometric Functions

Trig functions are periodic and definitely not one-to-one, so their inverses require carefully chosen domain restrictions.

Inverse Trig Domains and Ranges

• Arcsin ($\sin^{-1}$) has domain $[-1, 1]$ and range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$—these restrictions make sine one-to-one
• Arccos and arctan have their own specific restrictions—memorize these ranges, as they appear frequently on exams
• The restrictions aren't arbitrary: they're chosen to include all possible output values exactly once

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Calculus Connection: Derivatives of Inverses

This rule bridges Algebra 2 concepts with calculus—understanding it now gives you a head start.

The Reciprocal Derivative Rule

• The derivative of $f^{-1}(x)$ equals $\frac{1}{f'(f^{-1}(x))}$—the slopes of inverse functions are reciprocals at corresponding points
• Geometric meaning: where $f(x)$ is steep, $f^{-1}(x)$ is shallow, and vice versa
• Application: this lets you find tangent line slopes on inverse functions without explicitly solving for $f^{-1}(x)$

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Quick Reference Table

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Self-Check Questions

1. If $f(3) = 7$, what is $f^{-1}(7)$? What does this tell you about the point $(3, 7)$ on the graph of $f^{-1}$?

2. Why does $f(x) = x^2$ require a domain restriction to have an inverse, while $f(x) = x^3$ does not? Which test determines this?

3. Compare and contrast the graphs of $f(x) = 2x + 1$ and its inverse $f^{-1}(x) = \frac{x-1}{2}$. What line do they reflect over, and what happens to the point $(0, 1)$?

4. Given that $f(x) = 3x - 5$, use composition to verify that $f^{-1}(x) = \frac{x + 5}{3}$ is correct. What should $f(f^{-1}(x))$ equal?

5. If the domain of $f(x)$ is $[2, 8]$ and the range is $[-1, 5]$, state the domain and range of $f^{-1}(x)$. Explain why this swap occurs.