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hscscience Maths Adv · Y11
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Module 1 · L5 of 15 ~35 min ⚡ +50 XP in Learn · +25 to complete

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.

Today's hook, 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?
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Orient and recall

Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

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?

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Formula reference · this lesson
core notation
Even   $f(-x) = f(x)$  , symmetry: reflection in the $y$-axis
Odd   $f(-x) = -f(x)$  , symmetry: $180^\circ$ rotation about the origin
Neither   does not satisfy either condition

Key insight: The only function that is both even and odd is $f(x) = 0$.

03
What you'll master
Know

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
Understand

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
Can do

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
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Key terms
Function
A relation where each input has exactly one output.
Domain
The set of all possible input values for a function.
Range
The set of all possible output values for a function.
Even Function
A function where $f(-x) = f(x)$ for all $x$ in the domain; symmetric about the $y$-axis.
Odd Function
A function where $f(-x) = -f(x)$ for all $x$ in the domain; symmetric about the origin.
Neither
A function that is neither even nor odd.
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Even and odd functions

Work through the core explanation before applying it.

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Even and odd functions
core concept · +3 XP at end

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: $f(x) = x$, $f(x) = x^3$, $f(x) = x^5$, $f(x) = \sin(x)$

Neither even nor odd. Many functions satisfy neither condition. For example, $f(x) = x^2 + x$ has an even term ($x^2$) and an odd term ($x$). When tested, $f(-x) = x^2 - x$, which equals neither $f(x)$ nor $-f(x)$. Such functions have no special symmetry about the $y$-axis or the origin.

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.

Comparison table of even vs odd function symmetry

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

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Work examples end to end

Follow the reasoning through complete worked solutions.

Worked example 1 · testing for even or odd +5 XP on full reveal

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

1
Find $f(-x)$
Substitute $-x$ for every $x$ in the rule
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$f(-x) = (-x)^4 - 2(-x)^2 = x^4 - 2x^2$
Even powers of $-x$ are always positive, so signs don't change
3
Compare: $f(-x) = x^4 - 2x^2 = f(x)$ ✓
Since $f(-x) = f(x)$, the function is even
4
Check for oddness: $-f(x) = -(x^4 - 2x^2) = -x^4 + 2x^2 \neq f(-x)$
Answer: $f(x)$ is an even function; graph has $y$-axis symmetry.
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Worked example 2 · testing for odd +5 XP on full reveal

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

1
Find $f(-x) = (-x)^3 + (-x) = -x^3 - x$
Odd powers of $-x$ flip the sign of each term
2
Compare with $-f(x) = -(x^3 + x) = -x^3 - x$
Check whether $f(-x) = -f(x)$
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$f(-x) = -f(x)$ ✓, so the function is odd
Not even: $f(-x) \neq f(x)$. So the function is not even.
4
Answer: $f(x) = x^3 + x$ is an odd function ✓
Graph has $180°$ rotational symmetry about the origin
Step 1 of 4
Worked example 3 · neither even nor odd +5 XP on full reveal

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

1
Find $f(-x) = (-x)^2 - 3(-x) + 1 = x^2 + 3x + 1$
Substitute $-x$ carefully into every term
2
Compare with $f(x) = x^2 - 3x + 1$
The middle terms differ: $+3x$ vs $-3x$ → not equal → not even
3
Compare with $-f(x) = -(x^2 - 3x + 1) = -x^2 + 3x - 1$
$f(-x) = x^2 + 3x + 1 \neq -x^2 + 3x - 1$ → not odd
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Answer: $f(x)$ is neither even nor odd ✓
Any polynomial mixing even and odd powers is typically neither
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Common mistakes · the 4 traps that cost marks

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Dodge the traps, then apply

Meet the mistakes that cost marks, then do it yourself.

1

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 $-f(x)$.

2

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 $f(x) = 0$.

3

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."

4

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.

A

$f(x) = x^6$

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B

$f(x) = x^3 - x$

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C

$f(x) = x^2 + x + 1$

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D

$f(x) = \dfrac{1}{x}$

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Odd one out: Which of these functions is even? Select one.

Match each function to its type:

  • $f(x) = x^2$
  • $f(x) = x^3$
  • $f(x) = x^2 + x$
  • Odd, $f(-x) = -f(x)$
  • Neither even nor odd
  • Even, $f(-x) = f(x)$

Quick-fire practice · 5 reps +2 XP per reveal

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Drill it, then lock it in

Run the quick drill and copy the summary into your book.

1

Is $f(x) = x^6$ even, odd, or neither?

2

Is $f(x) = x^3 - x$ even, odd, or neither?

3

A function is even and $f(3) = 7$. What is $f(-3)$?

4

Is $f(x) = x^2 + x + 1$ even, odd, or neither?

5

Name the only function that is both even and odd.

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Revisit your thinking

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.

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Multiple choice

Answer the drill bank and rate your confidence.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

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Short answer

Write full responses, then check them against the model answers.

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Short answer
UnderstandBand 33 marks

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)

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ApplyBand 34 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)

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EvaluateBand 53 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)

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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.

Definition of an even function. $f(-x) = f(x)$ defines an even function.

Odd function test. $f(-x) = -f(x)$ for $f(x) = x^3$.

Symmetry of odd functions. $180°$ rotational symmetry about the origin.

Even powers. All powers are even; $f(-x) = f(x)$.

Confirming a function is odd. $f(-x) = -f(x)$, 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.

Module overview · Maths Advanced · Module quiz