Inverse Function Calculator
Finds f⁻¹(x) by undoing the operations in f(x) one at a time, from the outside in — the same method taught for finding an inverse by hand. Every result is checked by composing it back with the original function before it is shown.
How it works
How the Inverse Function Calculator works
the part a result on its own leaves out
What an inverse function undoes
If f takes an input and produces an output, f⁻¹ takes that output and returns the original input. Composing the two together in either order gives back exactly what you started with — f⁻¹(f(x)) = x, and f(f⁻¹(x)) = x.
Why the method is 'undo it, in reverse'
f(x) is built by applying operations to x in some order — multiply, then add, say. Its inverse undoes them one at a time, starting from the last operation applied and working back to the first. This only works when x appears exactly once; if it appears twice, as in (x+1)/(x-1), there is no single chain of operations to undo, and the calculator says so rather than guessing.
Common mistakes
Common mistakes with inverse functions
the errors that actually show up, not every possible slip
Undoing the operations in the original order instead of reverse
For f(x) = 2x + 3, the last thing done to x is add 3, so that is undone first: subtract 3, then divide by 2. Dividing first gives the wrong function entirely.
Forgetting a domain restriction on x² or sin(x)
x² is not invertible over all real numbers — both 2 and −2 square to 4. The usual inverse √x only undoes x² for x ≥ 0. The same issue applies to sin, cos and tan, whose inverses are only true inverses on a restricted principal domain.
Mixing up f⁻¹(x) with 1/f(x)
The −1 in f⁻¹ is not an exponent. f⁻¹(x) is the inverse function; 1/f(x) is the reciprocal, and the two are almost never the same function.
Questions
Asked often enough to answer here
answers that change how you work
How do I find the inverse of a function?
Write y = f(x), then undo each operation applied to x, working from the outside in — the last operation applied to x is the first one undone. Finally swap x and y so the input is named x again.
Why does the calculator decline some functions?
When x appears more than once, like in (x+1)/(x-1), there is no single chain of operations to undo — that needs solving an equation instead, which is out of scope here. The calculator says so rather than guessing.
What does f⁻¹(x) mean?
The inverse of f — a function that undoes what f does. It is not the same as 1/f(x), even though the notation looks similar.
Does every function have an inverse?
Only if it is one-to-one — no two inputs share an output. x² is not one-to-one over all real numbers, so its inverse √x is only valid once the domain is restricted to x ≥ 0.
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