3. Odd & Even Functions

Video coming soon3 worked examples

Theory

A function is even if $f(-x) = f(x)$ (symmetric about the $y$ -axis, like $x^2$ or $\cos x$ ), and odd if $f(-x) = -f(x)$ (rotational symmetry about the origin, like $x^3$ or $\sin x$ ). A function that satisfies neither is simply neither.

The Golden Rule: compute $f(-x)$ and compare: equal to $f(x)$ means even, equal to $-f(x)$ means odd, otherwise neither.

⚠️ Common Examiner Traps

A mixed function is neither: a sum like $x^2 + x$ is neither odd nor even — every term must fit the same pattern.
Negating the whole function: $-f(x)$ negates every term, not just some.
Even powers vs odd powers: even powers of $x$ are unchanged by $x \to -x$ ; odd powers change sign.

Worked examples

Example 1

Determine whether $f(x) = x^3 - 4x$ is odd, even, or neither.

Step 1: Compute $f(-x)$ :

f(-x) = (-x)^3 - 4(-x) = -x^3 + 4x

Step 2: Compare with $-f(x) = -(x^3 - 4x) = -x^3 + 4x$ . They match, so $f$ is odd.

Example 2

Determine whether $f(x) = x^4 + 3x^2 - 1$ is odd, even, or neither.

Step 1: Compute $f(-x)$ :

f(-x) = (-x)^4 + 3(-x)^2 - 1 = x^4 + 3x^2 - 1

Step 2: This equals $f(x)$ , so $f$ is even.

Example 3

Determine whether $f(x) = x^2 + x$ is odd, even, or neither.

Step 1: Compute $f(-x) = (-x)^2 + (-x) = x^2 - x$ .

Step 2: This is neither $f(x) = x^2 + x$ nor $-f(x) = -x^2 - x$ , so $f$ is neither odd nor even.