## Calculus: Early Transcendentals (2nd Edition)

$f(x)=x|x|$ Check for symmetry about the $y$-axis by substituting $x$ by $-x$ in the given function and simplifying: $f(-x)=-x|-x|=-x|x|$ Since substituting $x$ by $-x$ did not yield an equivalent expression, the function is not symmetric about the $y$-axis. Check for symmetry about the $x$-axis by substituting $f(x)$ by $-f(x)$ in the given function and simplifying: $-f(x)=x|x|$ Multiply both sides by $-1$: $f(x)=-x|x|$ Since substituting $f(x)$ by $-f(x)$ did not yield an equivalent expression, the function is not symmetric about the $y$-axis. Check for symmetry about the origin by substituting $x$ by $-x$ and $f(x)$ by $-f(x)$ and simplifying: $-f(x)=-x|-x|$ $-f(x)=-x|x|$ Multiply both sides by $-1$: $f(x)=x|x|$ Since the substitution yielded an equivalent expression, the function is symmetric about the origin. The graph is shown in the answer section