Proof of $f(f^{-1}(x))=x$ incomplete

Question

$$f(x)=y \tag 1$$ Swapping $x$ and $y$ creating an inverse function, $$f^{-1}(y)=x \tag 2$$ Substituting $(2)$ in $(1)$, $$f(f^{-1}(y))=y$$ But I want $x$ instead of $y$. Is it so that I can just replace $y$ with $x$ just like that, or What did I mistake?

Answer 1

There is no mistake, your result is correct. Your first equation $f(x)=y$ indicates that $x$ must lie in the domain of $f$, and $y$ in the range of $f$. What your final equation tells you is that $f(f^{-1}(y))=y$ for numbers $y$ which are in the range of $f$.

If your range and your domain are the same, you can simply rename your variables to get $f(f^{-1}(x))=x$.

Answer 2

If you say $g(y)=y$, you might tacitly mean that for all values of $y$ this equality holds, so you're saying $g(3)=3$ and $g(52)=52$ and $g(-7)=-7$ and so on. And that's the same as saying that for all values of $x$ you have $g(x)=x$. The variables $y$ and $x$ when used in this way are bound variables and so may be freely renamed. Google the terms "free variable" and "bound variable".