The given function is:
[tex]f(x)=-6x^5+4x^3[/tex]
Even functions are unchanged when reflected over the y-axis, so:
[tex]f(-x)=f(x)[/tex]
Odd functions are unchanged when rotated 180° about the origin, so:
[tex]f(-x)=-f(x)[/tex]
Now, replace -x as the argument of the function and let's observe the result:
[tex]\begin{gathered} f(-x)=-6(-x)^5+4(-x)^3 \\ f(-x)=-6*-x^5+4*-x^3 \\ f(-x)=6x^5-4x^3 \end{gathered}[/tex]
As can be observed, f(-x) is not equal to f(x), then this is not an even function.
Now, let's evaluate -f(x):
[tex]\begin{gathered} -f(x)=-(-6x^5+4x^3) \\ -f(x)=-(-6x^5)-(4x^3) \\ -f(x)=6x^5-4x^3 \\ THEN \\ f(-x)=-f(x) \end{gathered}[/tex]
This function is odd.
