Odd & Even Functions
Look at a butterfly's wings or the arch of the Sydney Harbour Bridge. Symmetry is everywhere in nature and design. In mathematics, special functions called even and odd functions capture this symmetry in precise, testable rules.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Consider the parabola $y = x^2$. What happens to the $y$-value when you replace $x$ with $-x$? Now consider $y = x^3$. Does the same thing happen? What kind of symmetry do you think each graph might have?
Key insight: The only function that is both even and odd is $f(x) = 0$.
Key facts
- The algebraic definitions of even and odd functions
- The geometric meanings of each type of symmetry
- That a function can be neither even nor odd
Concepts
- Why $f(-x) = f(x)$ corresponds to $y$-axis symmetry
- Why $f(-x) = -f(x)$ corresponds to origin symmetry
- How symmetry reduces the amount of working needed in analysis
Skills
- Algebraically test whether a function is even, odd, or neither
- Use symmetry properties to sketch graphs more efficiently
- Predict $f(-a)$ given $f(a)$ for even or odd functions
Some functions have special symmetry properties that make them easier to analyse and sketch. There are two main types: even functions and odd functions.
Even functions
A function is even if:
$$f(-x) = f(x)$$
This means replacing $x$ with $-x$ leaves the output unchanged. Geometrically, the graph of an even function is a mirror image across the $y$-axis.
Examples: $f(x) = x^2$, $f(x) = x^4$, $f(x) = |x|$, $f(x) = \cos(x)$
Odd functions
A function is odd if:
$$f(-x) = -f(x)$$
This means replacing $x$ with $-x$ flips the sign of the output. Geometrically, the graph of an odd function has rotational symmetry of $180^\circ$ about the origin. If you rotate the graph halfway around the point $(0, 0)$, it looks exactly the same.
Examples: $f(x) = x$, $f(x) = x^3$, $f(x) = x^5$, $f(x) = \sin(x)$
Misconceptions to fix
Wrong: A function is even if it has an even power somewhere in its rule.
Right: A function is even only if it satisfies $f(-x) = f(x)$ for all $x$ in its domain. You must test the whole rule, not just one term.
Wrong: Every function must be either even or odd.
Right: Many functions are neither even nor odd. If $f(-x)$ equals neither $f(x)$ nor $-f(x)$, state neither.
Even function: $f(-x) = f(x)$ — reflection in the $y$-axis — examples: $x^2$, $x^4$, $|x|$, $\cos(x)$; Odd function: $f(-x) = -f(x)$ — $180°$ rotational symmetry about the origin — examples: $x$, $x^3$, $\sin(x)$
Pause — copy the even function test ($f(-x) = f(x)$ ↔ $y$-axis symmetry) and the odd function test ($f(-x) = -f(x)$ ↔ 180° rotational symmetry) with examples into your book.
Did you get this? True or false: the function $f(x) = x^2 + x$ is an even function.
Quick check: Which statement best describes the geometric property of an odd function?
Worked examples · reveal as you go
Determine whether $f(x) = x^4 - 2x^2$ is even, odd, or neither.
Determine whether $f(x) = x^3 + x$ is even, odd, or neither.
Determine whether $f(x) = x^2 - 3x + 1$ is even, odd, or neither.
Common mistakes · the 4 traps that cost marks
Confusing even/odd with positive/negative coefficients
Some students think a function is "odd" because it has odd powers, or "even" because it has a positive leading coefficient. The names refer to the symmetry properties, not the sign of the coefficients.
✓ Fix: Always perform the algebraic test. Calculate $f(-x)$ and compare it to $f(x)$ and $-f(x)$.
Forgetting that the zero function is both even and odd
$f(x) = 0$ satisfies both $f(-x) = f(x)$ and $f(-x) = -f(x)$. It is the only function with this property.
✓ Fix: If asked for an example of a function that is both even and odd, the answer is $f(x) = 0$.
Assuming all functions must be even or odd
Most functions are neither. Any polynomial with both even and odd powers, or any function with a horizontal or vertical shift that breaks symmetry, will be neither.
✓ Fix: If $f(-x)$ equals neither $f(x)$ nor $-f(x)$, confidently state "neither."
Sign errors when computing $f(-x)$
For $f(x) = x^3 - 2x$, some students write $f(-x) = -x^3 - 2x$ instead of $-x^3 + 2x$. Each term must have its sign flipped individually.
✓ Fix: Substitute $-x$ into every term using brackets, then simplify each power separately.
Activity 1 — Even, odd, or neither?
For each function, determine whether it is even, odd, or neither. Show the algebraic test that justifies your answer.
$f(x) = x^6$
$f(x) = x^3 - x$
$f(x) = x^2 + x + 1$
$f(x) = \dfrac{1}{x}$
Odd one out: Which of these functions is even? Select one.
Match each function to its type:
Quick-fire practice · 5 reps +2 XP per reveal
Is $f(x) = x^6$ even, odd, or neither?
Is $f(x) = x^3 - x$ even, odd, or neither?
A function is even and $f(3) = 7$. What is $f(-3)$?
Is $f(x) = x^2 + x + 1$ even, odd, or neither?
Name the only function that is both even and odd.
Earlier you were asked: Consider $y = x^2$. What happens when you replace $x$ with $-x$? Now consider $y = x^3$. Does the same thing happen? What symmetry might each graph have?
For $y = x^2$, replacing $x$ with $-x$ gives $(-x)^2 = x^2$, so the output stays the same. This is the hallmark of an even function, and the graph has reflection symmetry in the $y$-axis. For $y = x^3$, replacing $x$ with $-x$ gives $(-x)^3 = -x^3$, so the output flips sign. This is the hallmark of an odd function, and the graph has $180^\circ$ rotational symmetry about the origin. Symmetry in functions is not just beautiful — it is a powerful tool that lets us predict behaviour, simplify calculations, and sketch graphs with far less effort.
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q1. Explain the difference between an even function and an odd function, both algebraically and geometrically. Use the functions $f(x) = x^2$ and $g(x) = x^3$ as examples in your explanation. (3 marks)
Q2. Determine whether each function is even, odd, or neither. Show the algebraic test for each. (a) $f(x) = 3x^4 - x^2$ (b) $f(x) = x^5 + 2x^3$ (c) $f(x) = x^2 - 2x + 3$ (4 marks)
Q3. A student claims that if a function contains only odd powers of $x$, it must be an odd function. Evaluate this claim. Is it always true? Provide a proof if it is true, or a counterexample if it is false. (3 marks)
Comprehensive answers (click to reveal)
Multiple choice — drill bank
MC answers and feedback are shown inline as you complete each question. Use the retry button to attempt a fresh set.
1. B — $f(-x) = f(x)$ defines an even function.
2. B — $f(-x) = -f(x)$ for $f(x) = x^3$.
3. C — $180°$ rotational symmetry about the origin.
4. A — All powers are even; $f(-x) = f(x)$.
5. B — $f(-x) = -f(x)$, so odd.
Activity 1 — Sort + classify model answers
A. $f(x) = x^6$ — Even. $f(-x) = (-x)^6 = x^6 = f(x)$.
B. $f(x) = x^3 - x$ — Odd. $f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x)$.
C. $f(x) = x^2 + x + 1$ — Neither. $f(-x) = x^2 - x + 1$, which equals neither $f(x)$ nor $-f(x)$.
D. $f(x) = \frac{1}{x}$ — Odd. $f(-x) = \frac{1}{-x} = -\frac{1}{x} = -f(x)$.
Short answer model answers
Q1 (3 marks): An even function satisfies $f(-x) = f(x)$ and has reflection symmetry in the $y$-axis [1]. For example, $f(x) = x^2$ gives $f(-x) = (-x)^2 = x^2 = f(x)$ [0.5]. An odd function satisfies $f(-x) = -f(x)$ and has $180°$ rotational symmetry about the origin [1]. For example, $g(x) = x^3$ gives $g(-x) = (-x)^3 = -x^3 = -g(x)$ [0.5].
Q2 (4 marks):
(a) $f(-x) = 3(-x)^4 - (-x)^2 = 3x^4 - x^2 = f(x)$ → Even
(b) $f(-x) = (-x)^5 + 2(-x)^3 = -x^5 - 2x^3 = -(x^5 + 2x^3) = -f(x)$ → Odd
(c) $f(-x) = (-x)^2 - 2(-x) + 3 = x^2 + 2x + 3$. Not equal to $f(x)$ or $-f(x)$ → Neither
Award 1 mark each for correct classification with working.
Q3 (3 marks): The student's claim is true [1]. If a function consists only of odd powers of $x$, each term satisfies $(-x)^{\text{odd}} = -x^{\text{odd}}$. When summed, $f(-x) = -f(x)$ [1–2]. For example, $f(x) = x^3 + 2x$ gives $f(-x) = -x^3 - 2x = -f(x)$. Therefore any polynomial with only odd powers is an odd function.
Five timed questions on odd and even functions. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
⚔ Enter the arenaClimb platforms, hit checkpoints, and answer symmetry questions. Quick recall, lighter than the boss.
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