What's the Difference Between an Odd Function and an Even Function?

An even function satisfies $f(-x)=f(x)$ and has symmetry about the $y$-axis. An odd function satisfies $f(-x)=-f(x)$ and has rotational symmetry of $180^\circ$ about the origin. Thus, the key difference is the effect of replacing $x$ with (-x).

How the Tests Work

For either classification, the domain must be symmetric about zero: whenever $x$ is included, (-x) must also be included. Substitute (-x) for $x$, simplify fully, then compare with $f(x)$ and (-f(x)). This algebraic test gives the same conclusion as the graph's symmetry.

For example, let $f(x)=x^2+2$. Then

$$f(-x)=(-x)^2+2=x^2+2=f(x).$$

Therefore, $f$ is even: changing the sign of $x$ does not change the output.

For $g(x)=x^3-4x$,

$$g(-x)=(-x)^3-4(-x)=-x^3+4x=-g(x).$$

Therefore, $g$ is odd because every output changes sign.

For $h(x)=x^2+x$, however, $h(-x)=x^2-x$, which equals neither nor (-h(x)). Therefore, is neither. The zero function, , is a special case: it is both odd and even on a domain symmetric about zero.

Common Misconception

The common misconception is that “odd” and “even” merely describe the powers in a formula. Classification depends on the entire function and its domain. Although $x^3+1$ contains an odd power, the constant term means it is neither odd nor even. Similarly, a restricted domain such as $x\geq 0$ usually prevents either classification because (-x) may not belong to the domain.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches HL, AHL 2.14 and may appear on Paper 1 or Paper 2. In a “verify” question, provide evidence that validates the classification; in a “show” question, give the required algebraic steps. Always write $f(-x)$, simplify it explicitly, compare it with $f(x)$ or (-f(x)), and state the corresponding symmetry. A graph-only claim is usually weaker than the exact substitution test. Use correct function notation throughout and finish with a clear conclusion.