All points on the line

y = x

except the point

(0, 0)

lie on the graph but the function is not defined at the origin.

From the equation we see that the graph is symmetrical about the line $y = x$ because interchanging $x$ and $y$ in the equation produces the same equation so that if $(a, b)$ satisfies the equation then $(b, a)$ also satisfies the equation.

For $x > 1$ each line $y = k$ for $k \leq e$ cuts the graph of $y = x^{1 / x}$ in two points $(a, a^{1 / a})$ and $(b, b^{1 / b})$ such that $a^{1 / a} = b^{1 / b}$ or, equivalently, $a^{b} = b^{a}$ . Hence the two points $(a, b)$ and $(b, a)$ are on the graph of $x^{y} = y^{x}$ . For example $2^{4} = 4^{2}$ so $(2, 4)$ and $(4, 2)$ are on the graph of $x^{y} = y^{x}$ .

The maximum point of the graph of $y = x^{1 / x}$ is the point $(e, e^{1 / e})$ so the point $(e, e)$ lies on the graph of $x^{y} = y^{x}$ .

Graph of y^x = x^y