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Calculus β

Inverse function

An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse.

Definition

A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f-1 ("inverse of f").

Notation

Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f-1(x)1/f(x).

Relevant observations

  • If g is the inverse function of f, then f is the inverse function of g.
  • The domain of f-1 is the range of f and the range of f-1 is the domain of f.
  • A function may not have an inverse function, but if it has, the inverse function is unique.