How to Find the Inverse of a Function: A Complete Guide (2026)

Understanding how to find the inverse of a function is one of the most important skills in algebra and pre-calculus. Whether you're a student preparing for an exam, a teacher building lesson plans, or simply someone brushing up on mathematics in 2026, this guide walks you through everything you need to know — with clear worked examples at every step.

Before diving into the steps, you can always verify your work using our free Inverse Function Calculator — just enter any function and get the inverse instantly.

What Is an Inverse Function?

Formally, if f(a) = b, then f⁻¹(b) = a. The inverse function "undoes" what the original function "does." For a function to have an inverse, it must be one-to-one (also called injective) — meaning each output corresponds to exactly one input.

The 3-Step Method to Find f⁻¹(x)

Step 1: Replace f(x) with y

Write the equation using y notation. This makes the algebra easier to follow and sets up the swap in Step 2.

Example: If f(x) = 2x + 3, write y = 2x + 3.

Step 2: Swap x and y

After swapping: x = 2y + 3

Step 3: Solve for y

Rearrange the equation to isolate y. This gives you f⁻¹(x).

x = 2y + 3
x − 3 = 2y
y = (x − 3) / 2

So f⁻¹(x) = (x − 3) / 2.

Finding the Inverse of Linear Functions

Linear functions of the form f(x) = ax + b (where a ≠ 0) always have inverses because they are always one-to-one. The inverse is:

f⁻¹(x) = (x − b) / a

This means the slope of the inverse is the reciprocal of the original slope, and the domain is all real numbers ℝ.

Inverse of Quadratic Functions

For f(x) = x², the function is not one-to-one over all reals (both x = 2 and x = −2 give f(x) = 4). By restricting the domain to x ≥ 0:

• Write: y = x²
• Swap: x = y²
• Solve: y = √x

Therefore f⁻¹(x) = √x with domain x ≥ 0.

Inverse of Exponential & Logarithmic Functions

Exponential and logarithmic functions are inverses of each other — a fundamental relationship in mathematics:

• f(x) = eˣ → f⁻¹(x) = ln(x) (domain x > 0)
• f(x) = ln(x) → f⁻¹(x) = eˣ (domain all ℝ)
• f(x) = 10ˣ → f⁻¹(x) = log₁₀(x) (domain x > 0)

How to Verify an Inverse Function

To confirm that g(x) is truly the inverse of f(x), check both compositions:

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

Example: Verify that f⁻¹(x) = (x − 3) / 2 for f(x) = 2x + 3:

f(f⁻¹(x)) = f((x−3)/2) = 2·(x−3)/2 + 3 = (x−3) + 3 = x ✓
f⁻¹(f(x)) = f⁻¹(2x+3) = (2x+3−3)/2 = 2x/2 = x ✓

Common Mistakes When Finding Inverse Functions

Here are the most frequent errors to avoid in 2026:

• Confusing f⁻¹(x) with 1/f(x): The notation f⁻¹ means the inverse function, NOT the reciprocal. For example, if f(x) = 2x, then f⁻¹(x) = x/2, not 1/(2x).
• Skipping domain restrictions: Always check whether the original function is one-to-one before finding the inverse.
• Not swapping both variables: You must replace every x with y and every y with x simultaneously.
• Forgetting to state the domain of f⁻¹: The domain of f⁻¹(x) equals the range of f(x).

Practice Problems

Test your understanding by finding the inverse of each function below. Use our Inverse Function Calculator to check your answers!

1. f(x) = 4x − 8
2. f(x) = x² (for x ≥ 0)
3. f(x) = x³
4. f(x) = eˣ + 1
5. f(x) = ln(2x)
6. f(x) = (x + 5) / 3

Conclusion

Finding the inverse of a function is a powerful algebraic skill that underpins much of higher mathematics. The three-step method — replace, swap, solve — works for virtually every function type when combined with proper domain analysis. Remember: always check that your function is one-to-one before attempting to find its inverse.

For instant results, use our free Inverse Function Calculator at the top of this site — it handles linear, polynomial, exponential, logarithmic, and trigonometric functions with full step-by-step solutions and interactive graphs. Bookmark it for your studies in 2026!

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