Inverse of a Function

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The inverse of a function \(f(x)\) is a function \(g(x)\) such that if \(f\) maps an element \('a'\) to an element \('b',\) \(g\) maps \('b'\) to \('a'.\)

In more formal terms, \(g\left( {f\left( x \right)} \right) = x.\) That is, \(g\) reverses the action of \(f\) on \(x.\)

An inverse cannot exist for every function. To see why, consider  \(f:\,\,A \to B\)  where \(\text{A}\) is the domain and \(\text{B}\) the co-domain. Consider the following maps:

(a)  This is an into map (into function). If we take the ‘inverse map’ (from \(\text{B}\) to \(\text{A,}\) using the same ‘links’ as in \(f\,\)), we see that the element \('g'\) cannot be assigned to any element in \(\text{A.}\) In other words, for an into function, some values in the co-domain are ‘left out’, and their ‘inverses’ do not exist.

\( \Rightarrow \)               We cannot defined an inverse for an into function.

(b)  This is a many-one map (a many-one function).

If we take the inverse map, the element \('e'\) will be assigned to two elements in \(\text{A,}\) that is, \('a'\) and \('b'.\)

Hence, the inverse map cannot be a function.

\( \Rightarrow \)               We cannot defined an inverse for a many-one function.

From this discussion, we conclude that for a function to be invertible, it should be one-one and onto (also called a bijective function).

As is intuitively clear, we can easily define an inverse for the map above.

\[\left( {{f^{ - 1}}(e) = a,\,\,{f^{ - 1}}(f) = b,\,\,{f^{ - 1}}(g) = c,\,\,{f^{ - 1}}(h) = d} \right).\]