# Function properties: parity and periodicity

> IB Mathematics Analysis and Approaches HL · Unit 2: Functions
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-properties-parity-and-periodicity/

This sub-topic covers core function properties of parity (even/odd symmetry) and periodicity (repeating behavior). You will learn to identify these properties algebraically and graphically, and use them to simplify problem-solving across calculus and graph sketching.

**Prerequisites:** [Function notation, domain and range](https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-notation-and-domain-range/); [Transformations of functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-transformations/)

## Learning objectives

- Identify even and odd functions algebraically and graphically
- Calculate the fundamental period of periodic functions, including transformed trigonometric functions
- Use parity and periodicity to simplify problem-solving and sketch graphs
- Apply these properties to solve exam-style calculus and graph questions

## Algebraic Definition of Parity

**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.

> **warning**
>
> Most functions are neither even nor odd! Always check against both conditions before concluding.

## Graphical Interpretation of Parity

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.

**Check your understanding**

Test your understanding:

1. If an even function passes through $(2, 5)$, what other point must it pass through?

   - $(-2, 5)$
   - $(2, -5)$
   - $(-2, -5)$
   - $(5, 2)$

   *Why:* Even functions are symmetric about the y-axis, so $(x,y)$ maps to $(-x, y)$.

## Periodicity and Fundamental Period

**Periodic Function** — 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.

*Notation:* $f(x + T) = f(x)$

*Example:* $f(x) = \sin(x)$ has a fundamental period of $2\pi$

For transformed trigonometric functions of the form $f(x) = A f(Bx + C) + D$:
- 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 $\sin(Bx + C)$, $\cos(Bx + C)$: Fundamental period $T = \frac{2\pi}{|B|}$
- For $\tan(Bx + C)$, $\cot(Bx + C)$: Fundamental period $T = \frac{\pi}{|B|}$

**Worked example:** Find the fundamental period of $f(x) = 3\cos\left(\frac{\pi x}{2} - \pi\right) + 1$.

1. Identify the coefficient of $x$, $B = \frac{\pi}{2}$. For cosine, we use the period formula $T = \frac{2\pi}{|B|}$.
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.

> **tip**
>
> IB exams almost always ask for the *fundamental period*, not any arbitrary period, so always give the smallest positive value.

> **Exam tip:** Always use the absolute value of B, since period is always positive.

## Common pitfalls

- **Wrong:** Assuming all functions are either even or odd.
  - Why it fails: Most functions do not satisfy either parity condition.
  - Correct: Always check $f(-x)$ against both $f(x)$ and $-f(x)$ before drawing a conclusion.
- **Wrong:** Skipping the domain symmetry check for parity.
  - Why it fails: If the domain is not symmetric about the origin (e.g. $f(x) = x^2$ for $x \geq 0$), parity cannot exist.
  - Correct: First confirm that for every $x$ in the domain, $-x$ is also in the domain before testing the function condition.
- **Wrong:** Forgetting the absolute value of B when calculating period.
  - Why it fails: Period is a positive quantity, and negative B does not change the magnitude of the period.
  - Correct: Always use $T = \frac{\text{base period}}{|B|}$ regardless of the sign of B.
- **Wrong:** Thinking the phase shift changes the period of a periodic function.
  - Why it fails: Phase shift only shifts the graph horizontally, it does not change how often the graph repeats.
  - Correct: Ignore the phase shift term (C) when calculating the period of a transformed trigonometric function.
- **Wrong:** Multiplying the base period by B instead of dividing.
  - Why it fails: Confusing horizontal stretch with the effect on period: increasing B compresses the graph horizontally, decreasing the period.
  - Correct: Always divide the base period by |B| to get the new period.

## 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 | $\sin x, T=2\pi$ |
| Transformed Sin/Cos | $T=\frac{2\pi}{\|B\|}$ | $A\sin(Bx+C)+D$ | $3\cos(2x), T=\pi$ |
| Transformed Tan | $T=\frac{\pi}{\|B\|}$ | $A\tan(Bx+C)+D$ | $\tan(\pi x), T=1$ |

## 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.

- [Solving function equations and inequalities](https://www.owlsprep.com/study/ib-math-aa-hl-u2-solving-function-equations-and-inequalities/)
- [Geometry & Trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-overview/)
- [Radians and the unit circle](https://www.owlsprep.com/study/ib-math-aa-hl-u3-radians-and-the-unit-circle/)

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