Inverse function

If a func­tion \(g\) is the in­verse of a func­tion \(f\), then \(g\) un­does \(f\), and \(f\) un­does \(g\). In other words, \(g(f(x)) = x\) and \(f(g(y)) = y\). An in­verse func­tion takes as its do­main the range of the origi­nal func­tion, and the range of the in­verse func­tion is the do­main of the origi­nal. To put that an­other way, if \(f\) maps \(A\) onto \(B\), then \(g\) maps \(B\) back onto \(A\). To in­di­cate the in­verse of a func­tion \(f\), we write \(f^{-1}\).

Examples

$$y=f(x) = x^3\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y^{1/3}$$
$$y=f(x) = e^x\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = ln(y)$$
$$y=f(x) = x+4\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y-4$$

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