Function properties: parity and periodicity

IB Mathematics Analysis and Approaches HL· AA HL 2.6· 45 min read

1. Algebraic Definition of Parity★★☆☆☆⏱ 15 min

📘 Definition

Parity

A property describing the symmetry of a function $f (x)$ when inputs are replaced by their negative, categorized as even or odd. Parity can only exist if the domain of the function is symmetric about the origin (if $x$ is in the domain, $- x$ is also in the domain).

Example:

$f (x) = x^{2}$ is even, $f (x) = x^{3}$ is odd, $f (x) = x + 1$ is neither

The two categories of parity follow simple algebraic conditions:

Even function: $f (- x) = f (x)$ for all $x$ in the domain
Odd function: $f (- x) = - f (x)$ for all $x$ in the domain

📐 Worked Example

Prove whether $f (x) = \frac{x ^{3} + x}{e ^{x^{2}} + 1}$ is even, odd, or neither.

1
First check the domain: $f (x)$ is defined for all real $x$ , which is symmetric about the origin, so parity is possible. Next calculate $f (- x)$ :
2
$f(-x) = \frac{(-x)^3 + (-x)}{e^{(-x)^2} + 1} = \frac{-x^3 - x}{e^{x^2} + 1}$
3
Factor out the negative sign from the numerator:
4
$f(-x) = -\left(\frac{x^3 + x}{e^{x^2} + 1}\right) = -f(x)$
5
Since $f (- x) = - f (x)$ , the function satisfies the condition for an odd function.

2. Graphical Interpretation of Parity★★☆☆☆⏱ 10 min

Parity directly corresponds to graphical symmetry, which allows you to quickly identify parity and sketch graphs faster.

Even functions are symmetric about the y-axis: reflecting the right half of the graph ( $x > 0$ ) over the y-axis gives the full graph.
Odd functions are symmetric about the origin: rotating the right half of the graph ( $x > 0$ ) 180° around the origin gives the full graph. For any point $(a, b)$ on an odd function, $(- a, - b)$ is also on the function.

📐 Worked Example

An odd function $f (x)$ passes through $(3, 12)$ . What other point must lie on the graph?

1
For an odd function, any point $(a, b)$ has a corresponding point $(- a, - b)$ .
2
Substitute $a = 3, b = 12$ : the corresponding point is $(- 3, - 12)$ .
3
Confirm: $f (- 3) = - f (3) = - 12$ , so $(- 3, - 12)$ is indeed on the graph.

✓ Quick check

Test your understanding:

If an even function passes through $(2, 5)$ , what other point must it pass through?
- $(- 2, 5)$
- $(2, - 5)$
- $(- 2, - 5)$
- $(5, 2)$
Reveal answer
$(-2, 5)$ —
Even functions are symmetric about the y-axis, so $(x, y)$ maps to $(- x, y)$ .

3. Periodicity and Fundamental Period★★★☆☆⏱ 20 min

📘 Definition

Periodic Function

$f (x + T) = f (x)$

A function is periodic if there exists a positive constant $T$ such that $f (x + T) = f (x)$ for all $x$ in the domain. The fundamental period (usually just called the period) is the smallest positive $T$ that satisfies this condition.

Example:

$f (x) = sin (x)$ has a fundamental period of $2 π$

For transformed trigonometric functions of the form $f (x) = A f (B x + C) + D$ :

The phase shift $C$ and vertical shift $D$ do not affect the period
The period depends only on the coefficient $B$ of $x$ inside the function

For $sin (B x + C)$ , $cos (B x + C)$ : Fundamental period $T = \frac{2 π}{∣ B ∣}$
For $tan (B x + C)$ , $cot (B x + C)$ : Fundamental period $T = \frac{π}{∣ B ∣}$

📐 Worked Example

Find the fundamental period of $f (x) = 3 cos (\frac{π x}{2} - π) + 1$ .

1
Identify the coefficient of $x$ , $B = \frac{π}{2}$ . For cosine, we use the period formula $T = \frac{2 π}{∣ B ∣}$ .
2
Substitute $B$ into the formula:
3
$T = \frac{2\pi}{\left|\frac{\pi}{2}\right|} = 2\pi \times \frac{2}{\pi} = 4$
4
Confirm that no smaller positive $T$ satisfies $f (x + T) = f (x)$ , so the fundamental period is 4.

Exam tip:

Always use the absolute value of B, since period is always positive.

4. Common Pitfalls

Wrong move:

Assuming all functions are either even or odd.

Why:

Most functions do not satisfy either parity condition.

Correct move:

Always check $f (- x)$ against both $f (x)$ and $- f (x)$ before drawing a conclusion.

Wrong move:

Skipping the domain symmetry check for parity.

Why:

If the domain is not symmetric about the origin (e.g. $f (x) = x^{2}$ for $x \geq 0$ ), parity cannot exist.

Correct move:

First confirm that for every $x$ in the domain, $- x$ is also in the domain before testing the function condition.

Wrong move:

Forgetting the absolute value of B when calculating period.

Why:

Period is a positive quantity, and negative B does not change the magnitude of the period.

Correct move:

Always use $T = \frac{base period}{∣ B ∣}$ regardless of the sign of B.

Wrong move:

Thinking the phase shift changes the period of a periodic function.

Why:

Phase shift only shifts the graph horizontally, it does not change how often the graph repeats.

Correct move:

Ignore the phase shift term (C) when calculating the period of a transformed trigonometric function.

Wrong move:

Multiplying the base period by B instead of dividing.

Why:

Confusing horizontal stretch with the effect on period: increasing B compresses the graph horizontally, decreasing the period.

Correct move:

Always divide the base period by |B| to get the new period.

5. Quick Reference Cheatsheet

Property	Condition	Key Feature	Example
Even Function	$f (- x) = f (x)$	Symmetric about y-axis	$x^{2}, cos x$
Odd Function	$f (- x) = - f (x)$	Symmetric about origin	$x^{3}, sin x$
Periodic Function	$f (x + T) = f (x)$	Repeats every $T$ units	$sin x, T = 2 π$
Transformed Sin/Cos	$T = \frac{2 π}{∣ B ∣}$	$A sin (B x + C) + D$	$3 cos (2 x), T = π$
Transformed Tan	$T = \frac{π}{∣ B ∣}$	$A tan (B x + C) + D$	$tan (π x), T = 1$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Identify parity of a given function
2022 · 2
Find period of transformed trig function
2023 · 1
Use parity to simplify integration

Going deeper

What's Next

Parity and periodicity are foundational properties that reappear across almost all remaining topics in IB AA HL. Parity simplifies the calculation of definite integrals over symmetric intervals, letting you avoid complex antiderivatives by eliminating odd terms from the integrand. Periodicity is core to working with trigonometric functions, which are central to differential equations, complex numbers, and vector calculus later in the course. Mastering these properties also speeds up graph sketching and multiple-choice questions on exam day, saving you time for longer, higher-mark problems.