Inverse function explorer
Check whether a function is one-to-one on its domain or on a restriction you choose, and solve for its exact inverse. When it is not one-to-one, the report gives two inputs with the same output.
Functions are limited to 1,000 characters, one variable, 24 nesting levels and 100 digits per number; the restriction is limited to 200 characters with exact numbers. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of round trips and an SVG plot. Use one variable, such as x, with ^ for powers. Up to 1,000 characters. Review the settings before running.
Your result
How to use Inverse function explorer
- Type the function, such as x^2 - 4x + 1.
- Optionally restrict the domain, such as x >= 2.
- Run the explorer and read the one-to-one result, the inverse formula and the domain of the inverse.
- Download the JSON report, the CSV of round trips or the plot.
Useful next steps
Functions are limited to 1,000 characters, one variable, 24 nesting levels and 100 digits per number; the restriction is limited to 200 characters with exact numbers. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of round trips and an SVG plot.
One-to-one test
For polynomial and rational functions the tool checks the sign of the derivative exactly on every piece of the domain and compares the images of the pieces. If the function turns around, it reports two inputs with the same output as a witness. For expressions where the variable appears once, each step such as exp, ln or sqrt is strictly monotone, and abs or even powers are checked for a sign change. Method: OpenStax, College Algebra 2e, section 3.7 Inverse Functions (2021).
Solving for the inverse
The equation y = f(x) is solved exactly when the supported form is linear or quadratic in x. A quadratic branch is emitted only when its sign is proved on every allowed domain cell. If different pieces need different branches, the report says branch-not-determined and leaves the formula empty. Sample round trips are evidence only. Single-occurrence expressions are inverted step by step, and the inverse domain is the exact supported range of f.
Limits of the method
Piecewise inverse formulas, cubic or higher formulas and expressions such as x*exp(x) or sin(x) are not written out. One-to-one status and inverse formula status are separate: an inverse can exist while this method cannot express it. The plot and the numerical round trips are evidence, not proof.