Question

Determine algebraically whether the given function is even, odd, or neither.

Answer 1

To check whether a function is odd or even, we simply substitute the argument by its negative version, namely "x" by "-x".

if the expression simplifies to resemble the original expression, that simply means the expression is even.  If it resembles the original negative expression, is odd.

$\bf g(x)=-2x^3-9\\\\ -------------------------------\\\\ \stackrel{x=-x}{g(-x)}=-2(-x)^3-9\implies g(-x)=-2(-x)(-x)(-x)-9 \\\\\\ g(-x)=+2(x)(x)(x)-9\implies g(-x)=2x^3-9$

 well, that doesn't look like the original - 2x³ - 9, so is not even.

and -f(x) would be 2x³ + 9, and that doesn't look like either, so is not odd.

thus is neither.