Inverse Function Calculator

The inverse of a function f, written f⁻¹, "undoes" what f does: if f(a) = b, then f⁻¹(b) = a. Finding an inverse algebraically means solving the equation y = f(x) for x in terms of y — straightforward for simple functions like f(x) = 2x + 3, but often difficult or impossible in closed form for more complex functions.

This calculator sidesteps that difficulty by solving f(x) = y numerically: it scans your specified search range for x, looking for where f(x) crosses your target value y, and refines the answer using the bisection method. This works for a very wide range of functions, including ones that don't have a simple algebraic inverse.

A function only has a true, well-defined inverse over its entire domain if it's one-to-one (injective) — meaning no two different inputs produce the same output. For example, f(x) = 2x + 3 is one-to-one everywhere, so it has a single, global inverse. But f(x) = x² is not one-to-one over all real numbers (both 2 and −2 map to 4), so "the" inverse only makes sense if you restrict the domain (e.g. to x ≥ 0, giving the inverse √y).

This calculator handles that reality by searching within a range you specify — narrowing the search range effectively lets you pick which branch of a multi-valued inverse you want.

For simple functions, finding the inverse algebraically (swapping x and y, then solving for y) is quick and gives an exact formula you can reuse for any input. But many functions — especially those mixing exponentials, logarithms, and polynomials, like f(x) = x + eˣ — have no algebraic inverse expressible with elementary functions at all.

A numeric approach like the one used here always works (given a valid search range with a root inside it), at the cost of only giving you the answer for one specific y value at a time, rather than a general formula.