Chapter F - Foundations: A Prelude to Functions - Section F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry - F.2 Assess Your Understanding - Page 17: 56

Intercepts: $y =0$ $x =0$ Origin symmetry.

To find the x-intercept of an equation, we set $y=0$ and solve: $y=\sqrt[5]{x}$ $0=\sqrt[5]{x}$ $x=0^5=0$ Thus, $x=0$ is the only x-intercept. Similarly, to find the y-intercept of an equation, we set $x=0$ and solve: $y=\sqrt[5]{x}$ $y=\sqrt[5]{0}=0$ Thus, $y=0$ is the only y-intercept. To check for symmetry about the x-axis, we replace every instance of $y$ with $-y$ in the original equation. The new equation must be equivalent to the original. We have: $-y= \sqrt[5]{x}$ which is not the same as $y= \sqrt[5]{x}$ Thus, there is no symmetry about the x-axis. For symmetry about the y-axis, replace every instance of $x$ with $-x$. We have $y = \sqrt[5]{-x}=-\sqrt[5]{x}$ which is not the same as $y = \sqrt[5]{x}$ Thus, there is no symmetry about the y-axis. However, the graph is symmetric with respect to the origin since if we replace $x,y$ by $-x,-y$, we get the same equation: $-y= \sqrt[5]{-x}= -\sqrt[5]{x}$ Thus $y= \sqrt[5]{x}$. which is the same as the original equation. Thus, we have origin symmetry.