Inverse Function Calculator

The inverse of f is the function f^-1 that undoes f: if y = f(x), then x = f^-1(y). On a graph the inverse is the reflection of f across the line y = x. The inverse only exists when f is one-to-one (each y comes from exactly one x). All nine families in this calculator are one-to-one on the stated domain, which is why the inverse is always well-defined for the inputs you can pick.

The full parabola y = a*x^2 + c is not one-to-one over all real x because both x and -x map to the same y. To get an inverse you have to restrict the domain to a single branch. The calculator picks the right half (x >= 0), so the inverse is x = sqrt((y - c)/a). If you need the left branch (x <= 0), take the negative of the result. Either branch by itself is one-to-one, but you cannot invert both at once.

y = (a*x + b) / (c*x + d) has an inverse exactly when a*d - b*c is non-zero. If the determinant is zero, the function collapses to a constant on its domain, which has no inverse. The inverse formula is x = (b - d*y) / (c*y - a) and it is itself a linear rational function. This is the family that includes Mobius (linear-fractional) transformations from complex analysis.

If you enter a test x, the calculator computes y = f(x) using the original formula, then x' = f^-1(y) using the inverse formula. For a correct inverse, x' should equal x within rounding. The check is a sanity test you can run on any input. If it fails, the most common cause is that your test x sits outside the family's domain (for example a negative x for the square-root family) or that floating-point precision broke down for very large or very small numbers.

Symbolic inversion is hard. Most functions you can write down have no closed-form inverse, and the ones that do often need case splits on the domain. Picking from nine well-behaved one-to-one families sidesteps the parsing problem entirely and guarantees you get a real formula back, not an error. For an arbitrary function, the next-best approach is numerical: pick a y, then solve f(x) = y with a root-finder. That is not what this calculator does.