why is $f^{-1} (x)$ , the reflection of $f(x)$ over the line $y=x$?

Question

I can understand the algebra but I just can not understand the intuition. For example consider $y=x^2$, I just don't understand how $x^2 =y$ is a reflection over the line $y=x$.

Answer 1

That is because, when you represent both graphs on the same figure, it amounts to swapping $x$ and $y$, and the mapping $(x,y)\longmapsto (y,x)$, geometrically, corresponds to the symmetry w.r.t. the first bissectrix of the coordinate axes (with equation $y=x$).

Indeed, the sum of the vectors with coordinates $(x,y)$ and $(y,x)$ is $(x+y, y+x)$, which lies on this bissectrix.