Answer
The function f is an odd function and is symmetric about the origin.
Work Step by Step
An odd function is the function having symmetry with the origin and an even function is the function having symmetry with the y-axis.
To check whether the function is even or odd, substitute $-x$ in the place of $x$ in the function and simplify as follows:
$\begin{align}
& f\left( -x \right)=2{{\left( -x \right)}^{3}}-6{{\left( -x \right)}^{5}} \\
& f\left( -x \right)=-2{{x}^{3}}+6{{x}^{5}} \\
& f\left( -x \right)=-\left( 2{{x}^{3}}-6{{x}^{5}} \right) \\
& f\left( -x \right)=-f\left( x \right)
\end{align}$
On putting –x in the place of x, it can be seen that same function with opposite signs is obtained. Thus the definition of an odd function is fulfilled and the graph of the function is symmetric about the origin.
Hence, the function is an odd function and the graph of the function is symmetric about the origin.
