Symmetry in $x=y$

Question

What are the properties of a function being symmetrical in $x=y$ ? how would one find the image of a function reflected in $x=y$ ? and what would one need to prove to show that two functions are symmetrical(images of one another) in $x=y$?

Answer 1

Hint: the first implication that I can think of is that the functional inverse would be equal to the function.

Answer 2

Interchanging $x$ and $y$ yields the same expression.

Answer 3

If a function's graph is symmetric with respect to the line $y = x$, then a point $(a, b)$ is on the graph if and only if the point $(b, a)$ is on the graph.

Example. $f(x) = \dfrac{1}{x}$

To find the image of a function reflected in the line $y = x$, interchange the values of $x$ and $y$.

Example. The image of the graph of $f(x) = \sqrt{x}$ is the graph of the function $g(x) = x^2, x \geq 0$.

To prove the graphs of $f(x)$ and $g(x)$ are symmetric with respect to the line $y = x$, show that $(a, b)$ is on the graph of $f(x)$ if and only if $(b, a)$ is on the graph of $g(x)$. Note that this implies that $f(x)$ and $g(x)$ are inverse functions.

Example. If $(a, b)$ lies on the graph of $f(x) = \sqrt{x}$, then $a = b^2$, so $(b, a)$ lies on the graph of $g(x) = x^2, x \geq 0$. If $(c, d)$ lies on the graph of $g(x)$, then $d = c^2$, with $c > 0$. Hence, $(d, c)$ lies on the graph of $f(x) = \sqrt{x}$.