How to Find the Inverse of a Function: Step-by-Step Guide for 2027

Inverse functions are the algebra topic students forget the fastest after the test. The reason is simple: most teachers introduce the algorithm (swap and solve) without explaining why it works. Once you see what an inverse actually does, the algorithm makes sense and the topic sticks.

This guide explains what an inverse is, gives you the four-step algebra recipe, covers the horizontal line test, and shows you the graphical and functional checks that confirm you found the right inverse.

What an Inverse Function Does

A function f takes an input x and produces an output y. The inverse function, written f⁻¹, undoes f: it takes that output back to the original input.

If f(3) = 7, then f⁻¹(7) = 3.

f⁻¹ is read “f inverse.” Important: the −1 here is not an exponent. f⁻¹(x) is not 1/f(x).

Notation

The last row is the definition of an inverse. Two functions are inverses if and only if composing them returns x.

When Does an Inverse Exist?

A function has an inverse if and only if it is one-to-one: each output corresponds to exactly one input.

Use the horizontal line test to check on a graph. If any horizontal line crosses the graph in more than one place, the function is not one-to-one, and it does not have an inverse over its full domain.

Examples: – f(x) = 3x − 5: passes the horizontal line test. Has an inverse. – f(x) = x²: fails. The horizontal line y = 4 crosses at x = 2 and x = −2. No inverse over all real numbers.

To give x² an inverse, restrict the domain to x ≥ 0. Then it passes the horizontal line test on that restricted domain, and the inverse is √x.

The Algorithm: Four Steps

To find the inverse of f(x):

1. Write y = f(x).
2. Swap x and y.
3. Solve for y.
4. Rename y as f⁻¹(x).

The swap step is what does the “undoing.” We are asking: given the original output, what input would have produced it?

Example 1: Linear Function

Find the inverse of f(x) = 3x − 5.

Step 1: y = 3x − 5. Step 2: x = 3y − 5. Step 3: x + 5 = 3y → y = (x + 5)/3. Step 4: f⁻¹(x) = (x + 5)/3.

Check: f(f⁻¹(x)) = 3 · (x + 5)/3 − 5 = (x + 5) − 5 = x. ✓

Example 2: Cubic Function

Find the inverse of f(x) = x³ + 1.

Step 1: y = x³ + 1. Step 2: x = y³ + 1. Step 3: x − 1 = y³ → y = ∛(x − 1). Step 4: f⁻¹(x) = ∛(x − 1).

Cube roots accept negative inputs, so the domain of f⁻¹ is all real numbers.

Example 3: Rational Function

Find the inverse of f(x) = (2x + 3) / (x − 1).

Step 1: y = (2x + 3)/(x − 1). Step 2: x = (2y + 3)/(y − 1). Step 3: Multiply both sides by (y − 1): x(y − 1) = 2y + 3. Distribute: xy − x = 2y + 3. Group y terms: xy − 2y = x + 3. Factor: y(x − 2) = x + 3. Divide: y = (x + 3)/(x − 2).

Step 4: f⁻¹(x) = (x + 3)/(x − 2).

Example 4: Function With Restricted Domain

Find the inverse of f(x) = x² − 4 for x ≥ 0.

Step 1: y = x² − 4. Step 2: x = y² − 4. Step 3: y² = x + 4 → y = ±√(x + 4).

Choose the positive branch because the original domain is x ≥ 0 (which means the original y values are ≥ −4 and the swap forces the inverse to have y ≥ 0).

Step 4: f⁻¹(x) = √(x + 4).

Example 5: Exponential and Logarithmic

Find the inverse of f(x) = 2^x.

Step 1: y = 2^x. Step 2: x = 2^y. Step 3: Take log base 2: y = log₂ x. Step 4: f⁻¹(x) = log₂ x.

This is the general fact: exponentials and logarithms are inverses of each other.

The Graph of an Inverse

The graph of f⁻¹ is the reflection of the graph of f across the line y = x. If (a, b) is on f, then (b, a) is on f⁻¹.

This means: – The x-intercept of f is the y-intercept of f⁻¹, and vice versa. – The domain of f is the range of f⁻¹. – The range of f is the domain of f⁻¹.

If you have a graph of f, you can sketch f⁻¹ by flipping each point across y = x without doing any algebra.

How to Verify Your Answer

Two valid checks:

1. Composition check. Compute f(f⁻¹(x)). If it simplifies to x, you have an inverse.
2. Point check. Pick a point on f, like (2, f(2)). Confirm that (f(2), 2) is on f⁻¹.

The composition check is the rigorous one. Do it on any inverse you find for a quiz or test.

Common Mistakes

1. Treating f⁻¹ as 1/f. They are not the same. f⁻¹(x) is the inverse function; 1/f(x) is the reciprocal.
2. Swapping but not solving for y. After the swap, x = 3y − 5 is not the answer. Isolate y.
3. Missing the ± when solving a quadratic. Pick the branch that matches the original domain restriction.
4. Forgetting that the inverse exists only on a restricted domain. Functions like x² need a restriction.
5. Confusing the inverse with the reflection. The graph of the inverse is the reflection across y = x, not the x-axis or the y-axis.

A Quick Cheat Sheet

Frequently Asked Questions

What are the steps to find the inverse of a function? There are four steps: write y = f(x), swap x and y, solve the new equation for y, and rename y as f⁻¹(x). For f(x) = 4x + 1: y = 4x + 1 becomes x = 4y + 1, so y = (x − 1)/4. Check by composing: f(f⁻¹(x)) = 4 × (x − 1)/4 + 1 = x.

What does finding the inverse of a function mean? Finding the inverse means finding a function that undoes the original one, taking each output back to its input. If f(2) = 9, then f⁻¹(9) = 2. For example, the inverse of converting Celsius to Fahrenheit, F = (9/5)C + 32, is C = (5/9)(F − 32). Note that f⁻¹(x) is not the same as 1/f(x).

How do you find the inverse of a function on a graph? Reflect the graph across the line y = x. Every point (a, b) on f becomes (b, a) on f⁻¹. If the graph passes through (0, 3), (1, 5) and (2, 7), the inverse passes through (3, 0), (5, 1) and (7, 2). Before reflecting, use the horizontal line test: if any horizontal line hits the graph twice, there is no inverse without restricting the domain.

How do you find the inverse of a function with a fraction? Swap x and y, then clear the fraction by multiplying both sides by the denominator, collect the y terms on one side and factor out y. For f(x) = (x + 2)/(x − 3): x = (y + 2)/(y − 3), so xy − 3x = y + 2, xy − y = 3x + 2, y(x − 1) = 3x + 2, and f⁻¹(x) = (3x + 2)/(x − 1).

Can you find the inverse of a function that is not one-to-one? Not over its whole domain, because one output would have to map back to two inputs. You can restrict the domain to a piece where the function is one-to-one. For f(x) = x², both 3 and −3 give 9, so there is no inverse on all real numbers, but restricting to x ≥ 0 gives the inverse f⁻¹(x) = √x.

How do you find the inverse of a function from a table? Swap the input and output columns. If a table lists x = 1, 2, 3, 4 with f(x) = 5, 8, 11, 14, the inverse table has x = 5, 8, 11, 14 with f⁻¹(x) = 1, 2, 3, 4. Check first that no output repeats; if two inputs share the same output, the inverse would not be a function.

Closing Thought

Finding an inverse is a four-step recipe (write, swap, solve, rename), and verifying it is a one-line composition check. Recognize when a function is not one-to-one, restrict the domain when needed, and sketch the reflection across y = x to double-check. Inverses are one of the cleanest topics in Algebra 2 once the algorithm clicks.

For more practice, browse our Algebra 2 worksheets and our full Math Topics library. When you are ready for a structured workbook, our Algebra 2 collection covers inverse functions in depth.