I have seen a lot of articles and books teaching how to find an inverse of a function by 'switching' $x$ and $y$ and solve for $y$ variables. My understanding of an inverse function is that it undoes whatever the function being given do, i.e. $f^{-1}(f(x))=x$. With this, I want to find the inverse of a function without using the 'switching of $x$ and $y$' since I can't find the reason why the 'switching of $x$ and $y$' works. Is there any other approach?
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1I totally agree with you. I use to say : "solve" for $x$ the equation $y=f(x)$ and $x=g(y)$ is the inverse. – Claude Leibovici Nov 21 '19 at 15:56