Editing Equation solving (section)

===Inverse functions===
{{See also|Inverse problem}}
In the simple case of a function of one variable, say, {{math|''h''(''x'')}}, we can solve an equation of the form {{math|''h''(''x'') {{=}} ''c''}} for some constant {{mvar|c}} by considering what is known as the ''[[inverse function]]'' of {{mvar|h}}.

Given a function {{math|''h'' : ''A'' → ''B''}}, the inverse function, denoted {{math|''h''<sup>−1</sup>}} and defined as {{math|''h''<sup>−1</sup> : ''B'' → ''A''}}, is a function such that

:<math>h^{-1}\bigl(h(x)\bigr) = h\bigl(h^{-1}(x)\bigr) = x \,.</math>

Now, if we apply the inverse function to both sides of {{math|''h''(''x'') {{=}} ''c''}}, where {{mvar|c}} is a constant value in {{mvar|B}}, we obtain

:<math>\begin{align}
h^{-1}\bigl(h(x)\bigr) &= h^{-1}(c) \\
x &= h^{-1}(c) \\
\end{align}</math>

and we have found the solution to the equation. However, depending on the function, the inverse may be difficult to be defined, or may not be a function on all of the set {{math|B}} (only on some subset), and have many values at some point.

If just one solution will do, instead of the full solution set, it is actually sufficient if only the functional identity

:<math>h\left(h^{-1}(x)\right) = x</math>

holds. For example, the [[projection (mathematics)|projection]] {{math|π<sub>1</sub> : '''R'''<sup>2</sup> → '''R'''}} defined by {{math|1=π<sub>1</sub>(''x'', ''y'') = ''x''}} has no post-inverse, but it has a pre-inverse {{math|π{{su|b=1|p=−1}}}} defined by {{math|1=π{{su|b=1|p=−1}}(''x'') = (''x'', 0)}}. Indeed, the equation {{math|π<sub>1</sub>(''x'', ''y'') {{=}} ''c''}} is solved by

:<math>(x,y) = \pi_1^{-1}(c) = (c,0).</math>

Examples of inverse functions include the [[nth root|{{mvar|n}}th root]] (inverse of {{math|''x''<sup>''n''</sup>}}); the [[logarithm]] (inverse of {{math|''a''<sup>''x''</sup>}}); the [[inverse trigonometric function]]s; and [[Lambert's W function|Lambert's {{mvar|W}} function]] (inverse of {{math|''xe''<sup>''x''</sup>}}).