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: f^{-1}(f(x))=f(f^{-1}(x))=x.\,

For example, if the function ''x'' → 3''x'' + 2 is given, then its inverse function is ''x'' → (''x''−2) / 3. This is usually written as:

: f\colon x o 3x+2
: f^{-1}\colon x o(x-2)/3

The superscript "−1" is not an exponent. Similarly, as long as we are not in s, e.g. arcsin ''x'' for the inverse of sin(''x'').

If a function ''f'' has an inverse then ''f'' is said to be invertible.

Simplifying rule

Generally, if ''f''(''x'') is any function, and ''g'' is its inverse, then ''g''(''f''(''x'')) = ''x'' and ''f''(''g''(''x'')) = ''x''. In other words, an inverse function undoes what the original function does. In the above example, we can prove ''f''−1 is the inverse by substituting (''x'' − 2) / 3 into ''f'', so
: 3(''x'' − 2) / 3 + 2 = ''x''.
Similarly this can be shown for substituting ''f'' into ''f''−1.

Indeed, an equivalent definition of an inverse function ''g'' of ''f'', is to require that ''g'' o ''f'' be the Identity Function on the Domain of ''f'', and ''f'' o ''g'' be the identity function on the Codomain of ''f'', where "o" represents Function Composition .


Existence

For a function ''f'' to have a valid inverse, it must be a Bijection , that is:
  • (''f'' is Onto ) each element in the Codomain must be "hit" by ''f'': otherwise there would be no way of defining the inverse of ''f'' for some elements.

  • (''f'' is One-to-one ) each element in the codomain must be "hit" by ''f'' only once: otherwise the inverse function would have to send that element back to more than one value.


If ''f'' is a real-valued function, then for ''f'' to have a valid inverse, it must pass the Horizontal Line Test , that is a horizontal line y=k placed on the graph of ''f'' must pass through ''f'' exactly once for all real ''k''.

It is possible to work around this condition, by redefining ''f'''s codomain to be precisely its Range , and by admitting a multi-valued function as an inverse.

If one represents the function ''f'' graphically in an ''x''-''y'' Coordinate System , then the graph of ''f'' −1 is the reflection of the graph of ''f'' across the line ''y'' = ''x''.

Algebraically, one computes the inverse function of ''f'' by solving the equation
: y=f(x)
for ''x'', and then exchanging ''y'' and ''x'' to get
:y=f^{-1}(x)
This is not always easy; if the function ''f''(''x'') is Analytic , the Lagrange Inversion Theorem may be used.

The symbol f −1 is also used for the (set valued) function associating to an element or a subset of the codomain, the Inverse Image of this subset (or element, seen as a Singleton ).


SEE ALSO