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 $y=x$.
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$f(x) = y$ if and only if $x = f^{-1}(y)$, so you're just switching the $x$ and $y$. Also, $f(x) = x^2$ has no inverse – user388557 Jun 13 '20 at 19:01
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2The reflection is not $x^2=y$ it’s $y^2=x$ – Ryan Shesler Jun 13 '20 at 19:03
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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.
Bernard
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