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Inverse Function Calculator

Find the Inverse

Use x as variable. Supported: linear, rational, sqrt, quadratic, power

Enter a function (e.g., 3x+2) and click "Find Inverse".

The calculator will show each algebraic step.

The Inverse Function Calculator finds the inverse of a given function step by step. Simply enter a function like 3x+2, (2x+1)/(x-3), or sqrt(2x-4) and the calculator will apply the swap‑and‑solve method, explaining every operation along the way.

How to Find the Inverse of a Function

The process is straightforward:

  1. Replace f(x) with y.
  2. Interchange x and y (this reflects the function over y = x).
  3. Solve the resulting equation for y.
  4. Replace y with f⁻¹(x).

Not all functions have inverses. A function must be one‑to‑one (each input gives a unique output). If it’s not one‑to‑one, you may restrict the domain (as we do for quadratic functions) to make it invertible.

Worked Examples

  • f(x) = 3x + 2 → f⁻¹(x) = (x – 2)/3
  • f(x) = (2x+1)/(x-3) → f⁻¹(x) = (3x+1)/(x-2)
  • f(x) = √(2x-4) → f⁻¹(x) = (x²+4)/2, x ≥ 0
  • f(x) = x² + 3 (x ≥ 0) → f⁻¹(x) = √(x-3), x ≥ 3
The Swap‑and‑Solve Method

The method of swapping x and y works because the inverse function “undoes” the original. If (a, b) lies on the graph of f, then (b, a) lies on the graph of f⁻¹. Swapping the coordinates is equivalent to reflecting over the line y = x.

For rational functions, solving for y may involve multiplying both sides and collecting like terms. Our calculator automates this algebra, so you can focus on understanding the concept.

Domain and Range of Inverse Functions

When you find the inverse of a function, the domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. For quadratic functions (e.g., f(x) = x² + 3), the inverse is not a function unless we restrict the original domain. Our calculator automatically includes domain restrictions in the result when necessary (e.g., x ≥ 0 for the principal square root branch).

Understanding inverse functions is crucial in calculus (for derivatives of inverse functions), algebra (solving equations), and applied mathematics (decoding formulas). Use this calculator to check your homework or explore how different functions invert.

The calculator supports linear functions (ax + b), rational functions of the form (ax+b)/(cx+d), square roots √(ax+b), quadratics x² + c, and basic powers xⁿ. For other forms, the algebra can be more complex, but the swap‑and‑solve principle remains the same.

Frequently Asked Questions about Inverse Functions

What is an inverse function?
An inverse function reverses the operation of the original function. If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse is the reflection of the original function over the line y = x.
How do you find the inverse of a function?
Step 1: Replace f(x) with y. Step 2: Swap x and y. Step 3: Solve the new equation for y. Step 4: Replace y with f⁻¹(x).
When does a function have an inverse?
A function has an inverse if it is one‑to‑one (bijective), meaning each output corresponds to exactly one input. Graphically, it passes the horizontal line test.
What about functions like x²?
x² is not one‑to‑one over all reals because both 2 and –2 give 4. However, if we restrict the domain (e.g., x ≥ 0), then its inverse is √x.
Does the calculator handle all functions?
This version supports linear, simple rational, square root, quadratic (with domain restriction), and basic power functions. More complex functions may require advanced algebra.