How do you tell if a function is odd or even?

Replace x by −x and simplify. If you get f(x) back, the function is even; if you get −f(x), it is odd; otherwise it is neither. On a graph, even means mirror symmetry in the y-axis and odd means half-turn symmetry about the origin.

Are rules like "odd + odd = odd" examined?

The curriculum notes say such properties are not required. In practice you just test each term of the integrand with f(−x).

Is the absolute value function even?

Yes. |−x| = |x|, so y = |x| is even and its graph is a V symmetric about the y-axis.

Odd and Even Functions (DSE M2)

A function is even if f(−x) = f(x) (graph symmetric about the y-axis) and odd if f(−x) = −f(x) (graph symmetric about the origin). In M2 the point of the topic is the definite integral over [−a, a]: it is 0 for an odd function and 2∫₀^a f(x) dx for an even one.

Solve Odd and Even Functions questions with AI

The odd function y = x³ − x from Example 1 (blue) and the even function y = x² − 1 (red). Each marked pair of points shows the symmetry. Try it yourself →

Odd and Even Functions: Key Facts

Evenf(−x) = f(x): e.g. x², cos x, |x|

Oddf(−x) = −f(x): e.g. x³, sin x, tan x

Integral of an odd function∫_−a^a f(x) dx = 0

Integral of an even function∫_−a^a f(x) dx = 2∫₀^a f(x) dx

Absolute value|x| = x if x ≥ 0, −x if x < 0 (an even function)

Worked Examples — Odd and Even Functions

Example 1: Testing whether a function is odd or even

Determine whether f(x) = x³ − x is odd, even or neither.

f(−x) = (−x)³ − (−x) = −x³ + x
= −(x³ − x) = −f(x)

Answer: f is an odd function.

Example 2: An integral with an absolute value

Evaluate ∫₋₃³ x|x| dx.

Let g(x) = x|x|. Then g(−x) = (−x)|−x| = −x|x| = −g(x)
g is odd and the limits are −3 and 3

Answer: 0

💡 Exam Traps & Secrets

⚠️ Most functions are neitherf(x) = x² + x is neither odd nor even. Always compute f(−x) and compare it with both f(x) and −f(x) before using a symmetry shortcut.

⚠️ The shortcut needs symmetric limitsThe results hold only for ∫_−a^a. An odd function integrated from −1 to 2 is not 0.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Odd × odd)Is f(x) = x sin x odd, even or neither? Show it on a graph.

(Splitting into odd and even parts)Evaluate the integral of (x³ + 2x² − x) from −2 to 2.

(Absolute value graph)Sketch y = |x − 1| and state whether it is an even function.