Inverse function

The inverse function \(f^{-1}\), or inverse, of a function \(f\), is the function that undoes the operation of \(f.\) It exists as a well-defined function if and only if \(f\) is bijective.

Recall that the graph of a function \(f: X \rightarrow Y\) is a subset of \(X \times Y.\) The graph of its inverse \(f^{-1}\) has all the same ordered pairs \((x, y)\), but the roles of \(x\) and \(y\) are swapped. Therefore, the graph of \(f^{-1}\) is: $$f^{-1} = \set{(y, x) : (x, y) \in f}$$

The arrow diagram of an inverse function is the same as the arrow diagram as the original function, with one exception: all the arrows are reversed.

In function notation, the inverse of \(f: X \rightarrow Y\) is such that for all \(x \in X\) and \(y \in Y\), \(f(x) = y\) if and only if \(f^{-1}(y) = x.\) From this definition, you can deduce that \(f^{-1}(f(x)) = x\), the inverse undoes the original function.

Given a bijective function \(f: X \rightarrow Y\), where \(f(x) = y\), you can solve for its inverse by isolating \(x\) to find \(f^{-1}(y) = x.\)

⚠ Worrying about the use of the variable \(x\) or \(y\) in an inverse function is not necessary, because \(f^{-1}(y)\) and \(f^{-1}(x)\) are the same function. What matters is the operation \(f^{-1}\) applies to an input to undo \(f.\)
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