Q.Determine whether each function is even, odd, or neither:

Replace $x$ by $-x$, simplify, and compare the result with $f(x)$ (even) and with $-f(x)$ (odd).

(i) $f(x)=x^4-3x^2$. Compute $f(-x)$:

Since $f(-x)=f(x)$ for all $x$, $f$ is even.

(ii) $g(x)=x^3+2x$. Compute $g(-x)$:

Since $g(-x)=-g(x)$ for all $x$, $g$ is odd (and consistently $g(0)=0$).

(iii) $h(x)=x^2+x$. Compute $h(-x)$:

Compare: $h(x)=x^2+x$ and $-h(x)=-x^2-x$. Since $x^2-x$ equals neither of these (they differ in the sign of one term), $h$ is neither even nor odd. …

A quick guide for polynomials: a polynomial in which every term has an even power (including a constant term) is even; one in which every term has an odd power is odd; a mix of even and odd powers (like $x^2+x$) is neither. This shor …

Students often mislabel any polynomial with a highest even power as 'even' without checking the lower terms — the presence of an odd-power term like $+x$ in …