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Find and Write an
Inverse Function

Learn how to reverse any function: swap inputs and outputs, solve for the new variable, and verify with composition.

Algebra 2

What Is an Inverse Function?

A function \( f \) takes an input \( x \) and produces an output \( y \). Its inverse, written \( f^{-1} \), reverses that process: it takes the output \( y \) and returns the original input \( x \).

Think of a function as a machine: you feed in a number, and it transforms it. The inverse is the machine running in reverse: feed in the result and recover the original number.

Mapping Forward and Backward

Domain Range 1 2 3 3 5 7
\( f(x) = 2x + 1 \)
\( f^{-1}(x) \)

Why Do Inverses Matter?

Solving Equations
To solve \( f(x) = c \), apply \( f^{-1} \) to both sides and get \( x = f^{-1}(c) \).
Real-World Models
Convert Celsius to Fahrenheit and back, decode encrypted messages, or reverse engineering formulas.
Calculus Prep
Logarithms are the inverse of exponentials. Arcsine is the inverse of sine. Inverses are everywhere.

The One-to-One Requirement

A function has an inverse only if it is one-to-one (injective): every output is produced by exactly one input. Graphically, this means it passes the Horizontal Line Test.

✓ One-to-One 1 hit ✕ Not One-to-One 2 hits

Core Rule: Finding the Inverse

1
Replace \( f(x) \) with \( y \). This makes the algebra easier to manipulate.
2
Swap \( x \) and \( y \). This reflects the function across the line \( y = x \).
3
Solve for \( y \). Isolate the new \( y \) using algebraic operations.
4
Write the result as \( f^{-1}(x) \). Replace \( y \) with inverse notation.
5
Verify: check that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Graphical View: Reflection Across \( y = x \)

\( y = x \)
\( f(x) = 2x + 1 \)
\( f^{-1}(x) \)
 x y 1 2 3 -1 -1 1 2

Inverse vs. Reciprocal: Key Distinction

Inverse Function \( f^{-1}(x) \) Reciprocal \( \frac{1}{f(x)} \)
Notation \( f^{-1}(x) \) \( [f(x)]^{-1} \)
Meaning Undoes the function One divided by the output
Example If \( f(x) = 2x \), then \( f^{-1}(x) = \frac{x}{2} \) If \( f(x) = 2x \), then \( \frac{1}{f(x)} = \frac{1}{2x} \)
Key Test \( f(f^{-1}(x)) = x \) \( f(x) \cdot \frac{1}{f(x)} = 1 \)

How to Apply the Steps

Follow this checklist every time you find an inverse:

1
Confirm one-to-one. Check with the Horizontal Line Test or verify algebraically.
2
Write \( y = f(x) \). Replace function notation with \( y \).
3
Swap \( x \leftrightarrow y \). Every \( x \) becomes \( y \) and vice versa.
4
Solve for \( y \). Use inverse operations: if \( f \) multiplied, divide; if \( f \) cubed, take a cube root.
5
Verify. Compose \( f(f^{-1}(x)) \) and confirm it equals \( x \).

Example 1: Linear Function

1
Write: \( y = 3x - 7 \)
2
Swap: \( x = 3y - 7 \)
3
Solve: \( x + 7 = 3y \), then \( y = \frac{x + 7}{3} \)
4
Result: \( f^{-1}(x) = \frac{x + 7}{3} \)

Example 2: Cubic Function

1
Write: \( y = 2x^3 \)
2
Swap: \( x = 2y^3 \)
3
Solve: \( \frac{x}{2} = y^3 \), then \( y = \sqrt[3]{\frac{x}{2}} \)
4
Result: \( f^{-1}(x) = \sqrt[3]{\frac{x}{2}} \)

Example 3: Rational Function

1
Write: \( y = \frac{x+1}{x-1} \)
2
Swap: \( x = \frac{y+1}{y-1} \)
3
Solve: \( x(y-1) = y+1 \) → \( xy - x = y + 1 \) → \( xy - y = x + 1 \) → \( y(x-1) = x+1 \)
4
Result: \( f^{-1}(x) = \frac{x+1}{x-1} \)

Verifying with Composition

Two functions \( f \) and \( g \) are inverses of each other if and only if both compositions return \( x \):

Test 1
\( f(g(x)) = x \) for all \( x \) in the domain of \( g \)
Test 2
\( g(f(x)) = x \) for all \( x \) in the domain of \( f \)

Domain and Range Swap

When you find the inverse, the domain of \( f \) becomes the range of \( f^{-1} \), and vice versa. Always state the domain of the inverse.

\( f(x) \) \( f^{-1}(x) \)
Domain All \( x \) where \( f \) is defined Range of \( f \)
Range All outputs of \( f \) Domain of \( f \)
Example \( f(x) = x^2,\ x \geq 0 \)
Domain: \( [0, \infty) \), Range: \( [0, \infty) \)		 \( f^{-1}(x) = \sqrt{x} \)
Domain: \( [0, \infty) \), Range: \( [0, \infty) \)

Common Mistakes to Avoid

✕ Confusing Inverse and Reciprocal
Writing \( f^{-1}(x) = \frac{1}{f(x)} \) instead of finding the actual inverse function.
✕ Forgetting to Swap
Solving for \( x \) in terms of \( y \) but not swapping \( x \) and \( y \) first (or at the end).
✕ Ignoring Domain Restrictions
Not restricting the domain when the original function is not one-to-one (e.g., \( x^2 \) needs \( x \geq 0 \)).

Question 1 of 5

What is the inverse of \( f(x) = 5x + 3 \)?

Question 2 of 5

If \( f(x) = 2x^3 \), what is \( f^{-1}(16) \)?

Question 3 of 5

Which condition must a function satisfy to have an inverse?

Question 4 of 5

What is the inverse of \( f(x) = \frac{x + 1}{x - 1} \) for \( x \neq 1 \)?

Question 5 of 5

If \( f \) and \( g \) are inverse functions, which must be true?