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Inverse Function Calculator
Last updated: May 31, 2026
Written by Blake Boege
An inverse function calculator is an algebraic solver designed to find the analytical inverse f⁻¹(x) of a one-to-one mathematical function f(x). It replaces f(x) with y, swaps the variables x and y, and solves the resulting equation to isolate the new dependent variable. It supports common function forms, including linear equations, rational functions, and power terms.
Find the algebraic inverse function f⁻¹(x) of linear, rational, power, and root expressions step-by-step.
Quick Answer
Solve for the inverse function f⁻¹(x) of a given math function f(x). Shows step-by-step algebraic variable swapping.
Mathematical Function
Input common functions of x like 3x - 7, (2x+1)/(x-3), or x^3 + 2.
e.g. 2x + 5
Use a single letter matching the function variable. · e.g. x
Inverse f⁻¹(x)
(x - 5) / 2
An inverse function f⁻¹(x) exists only if f(x) is one-to-one over its domain (i.e. passes the horizontal line test).
Examples
Linear f(x) = 2x + 5
f⁻¹(x) = (x − 5) ÷ 2
Rational f(x) = (x + 2) ÷ (x − 3)
f⁻¹(x) = (3x + 2) ÷ (x − 1)
How it works
The calculator parses the function and automatically solves for the independent variable, flipping coordinates to output f⁻¹(x).
Linear Inverse · (ax + b) → (x − b)/a
Rational Inverse · (ax + b)/(cx + d) → (d·x − b)/(a − c·x)
Algebraic steps to find the inverse
To invert a function f(x) = y:
- Set y = f(x): Write down the equation with y instead of f(x).
- Swap variables: Rewrite the equation, replacing every instance of y with x, and x with y.
- Solve for y: Isolate the new variable y on one side of the equation.
- Substitute notation: Declare the final equation as f⁻¹(x).
Related calculus and algebra calculators
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- Derivative Calculator — calculate the derivative or rate of change of any mathematical function.
- Domain Calculator — determine the set of allowable x inputs for functions.
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Frequently asked questions
An inverse function, denoted as f⁻¹(x), is a function that 'undoes' the action of the original function f(x). If f(x) = y, then f⁻¹(y) = x.
To find the inverse: 1) Replace f(x) with y. 2) Swap x and y in the equation. 3) Solve the new equation for y. 4) Replace y with f⁻¹(x).
No. A function only has an inverse if it is one-to-one (injective), meaning that for every output y, there is exactly one input x. Visually, the function's graph must pass the horizontal line test.
The graph of f⁻¹(x) is a reflection of the graph of f(x) across the diagonal line y = x. This means that if (a, b) is on the graph of f(x), then (b, a) is on the graph of f⁻¹(x).
You verify by composing them in both directions. If f(g(x)) = x and g(f(x)) = x, then f(x) and g(x) are inverses of each other.
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