If we consider the general \(3^{rd}\) degree polynomial function,

then

for any values of \(a\), \(b\), \(c\), or \(d\) since \(a \neq 0\).

Therefore, a cubic function is never an even function.

Now, \(-f(x)=-ax^3-bx^2-cx-d\),

so \(f(-x)=-f(x)\) when \(b=0\) and \(d=0\), that is when \(f(x)=ax^3+cx\).

Therefore, cubic functions of the form \(f(x)=ax^3+cx\), \(a \neq 0\), are odd functions.

Similarly, it should follow that even polynomial functions would have only even degree terms.

If we consider the general \(4^{th}\) degree polynomial function,

then

Setting the coefficients \(b=0\) and \(d=0\) will result in \(f(-x)=f(x)\), and hence an even function.

Therefore, quartic functions of the form \(f(x)=ax^4+cx^2+e\), \(a \neq 0\), are even functions.

And,

for any values of \(a\), \(b\), \(c\), or \(d\) since \(a \neq 0\).

Therefore, a quartic function is never an odd function.

To illustrate the results graphically, we compare the following functions: