Consider we have an function $f(x)$ and it has a inverse relation and consider that function starts from a point somewhere else in $+x$ axis. When we want to draw its $f^{-1}(x)$ (inverse function) we are making his symmetry based on $x=y$ because its changes its domain set and $(x)$ codomain $(y)$ so, when we want to get a symmetry of $+x$ its being $+y$ because we get its symmetry based on $x=y$.
Is every function manipulation based on making symmetry work likes this, and is this true to think like this?