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Calculator Answers

Inverse Function Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

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 Function Result

Inverse f⁻¹(x)

(x - 5) / 2

Inversion Process Stepsf(x) = 2x + 5 Step 1: Set y = f(x) y = 2x + 5 Step 2: Swap variables x and y x = 2y + 5 Step 3: Solve for y x - 5 = 2y y = (x - 5) / 2 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).

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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:

  1. Set y = f(x): Write down the equation with y instead of f(x).
  2. Swap variables: Rewrite the equation, replacing every instance of y with x, and x with y.
  3. Solve for y: Isolate the new variable y on one side of the equation.
  4. Substitute notation: Declare the final equation as f⁻¹(x).

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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.