Classify functions as even, odd, or neither

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Compare outputs at opposite inputs using the domain, rule, table, or graph

A function gives one output for each input in its domain. To classify a function, compare its outputs at opposite inputs, such as

3

and

-3

. First check the domain: it must contain the opposite of every input for the standard even or odd classification. Then check whether opposite inputs give equal outputs or outputs that are opposites. This lesson shows how to make those checks and decide when a function is neither.

What you will learn

State the defining tests for even and odd functions.
Check whether a function's domain is symmetric about zero.
Classify a function as even, odd, or neither using its rule, a table, or a graph.

1. Prerequisite bridge: inputs, outputs, and domain

The domain is the set of inputs for which a function is defined. In

f(x)

,

x

represents an input. The expression

f(-x)

means to replace every

x

in the function rule with

-x

.

Opposite inputs have the same distance from zero but different signs. For example,

4

and

-4

are opposite inputs. Their outputs are

f(4)

and

f(-4)

.

A domain is symmetric about zero if it contains the opposite of each of its inputs. The real numbers are symmetric about zero. A domain containing only

1

is not, because it does not contain

-1

. In symbols, the domain condition is: if

x

belongs to the domain, then

-x

belongs to it as well.

x∈ D\implies -x∈ D

Check the domain before applying the even or odd tests.
The standard even and odd classifications require a domain symmetric about zero.

2. Plain-language rules and visual patterns

An even function has equal outputs at opposite inputs. Its graph is symmetric about the

y

-axis: reflecting the graph across that axis leaves it unchanged. For example, if the graph contains

(3,5)

, it also contains

(-3,5)

.

An odd function has opposite outputs at opposite inputs. Its graph is symmetric about the origin: turning the graph halfway around the origin leaves it unchanged. If the graph contains

(3,5)

, it also contains

(-3,-5)

. If zero is in the domain of an odd function, its output at zero must be zero.

A function is neither when it satisfies neither definition. If the domain is symmetric about zero, compare outputs across the domain, not only at one pair. A single pair that fails a relationship rules out that classification.

If the domain is not symmetric about zero, it does not meet the standard domain requirement for an even or odd function. Do not use missing opposite-input comparisons as evidence that a function is even or odd.

f(-x)=f(x) or f(-x)=-f(x)

Even means equal outputs at opposite inputs.
Odd means opposite outputs at opposite inputs.
Neither means the function meets neither relationship.

3. Check with numbers, a rule, or a graph

A table can show what happens at listed opposite inputs. Equal outputs at a pair fit the even pattern; outputs that are opposites fit the odd pattern. If neither relationship holds for a pair, that pair rules out both classifications. A short table cannot establish a pattern for every input by itself.

For a formula, replace every

x

with

-x

and simplify. Compare the result with the original function,

f(x)

, and its opposite,

-f(x)

. Confirm that the domain is symmetric about zero before making the final classification.

For example, consider

f(x)=x^2+1

. Its domain is all real numbers, so it is symmetric about zero. Replacing

x

with

-x

gives

(-x)^2+1=x^2+1

, which matches

f(x)

. This function is even.

A graph can suggest the same patterns. Reflection across the

y

-axis suggests evenness; symmetry about the origin suggests oddness. Check the whole graph and its domain, rather than just one visible section.

f(-x)=f(x) or f(-x)=-f(x)

Substitute $-x$ everywhere in the rule, then simplify.
Compare $f(-x)$ with both $f(x)$ and $-f(x)$ .
Use the rule or the full graph to check the relationship across the domain.

Patterns at opposite inputs

Classification	Output relationship	Graph pattern
Even	$f(-x)=f(x)$	Symmetry about the $y$ -axis
Odd	$f(-x)=-f(x)$	Symmetry about the origin
Neither	Neither relationship holds across the domain	Neither symmetry pattern

Worked example

Classify a function using its rule

Classify

g(x)=\frac{x}{x^2+4}

as even, odd, or neither.

Check the domain
The denominator is positive for every real input, so the function is defined for all real numbers. Its domain is symmetric about zero.
Substitute the opposite input
Replace every occurrence of $x$ with $-x$ .
$g(-x)=\frac{-x}{(-x)^2+4}$
Simplify and compare
Since $(-x)^2=x^2$ , the denominator stays the same and the numerator changes sign. The result is the opposite of the original function.
$g(-x)=\frac{-x}{x^2+4}=-g(x)$
Classify
The odd-function relationship holds for every real input, and the domain is symmetric about zero. Therefore, $g$ is odd.

Answer:

g(x)=\frac{x}{x^2+4}

is odd.

Check: For example,

g(2)=\frac{1}{4}

and

g(-2)=-\frac{1}{4}

. These outputs are opposites, as the rule test predicts.

Common mistakes and how to avoid them

Calling a function even or odd after checking only one pair of inputs.

Correction: One pair can rule out a classification if it fails the required relationship, but a match at one pair does not establish the relationship for every input. Use the rule or full graph.

Replacing

x

with

-x

in only part of the rule.

Correction: Replace every occurrence of

x

, including in exponents, numerators, and denominators.

Ignoring the domain when classifying a function.

Correction: First check that the domain contains the opposite of each input. The standard even and odd classifications require a domain symmetric about zero.

Calling a function odd because one of its outputs is negative.

Correction: Oddness is about the relationship between outputs at opposite inputs. Check whether

f(-x)=-f(x)

throughout the domain.

Lesson summary

The standard even and odd definitions require a domain symmetric about zero.
An even function satisfies $f(-x)=f(x)$ across its domain.
An odd function satisfies $f(-x)=-f(x)$ across its domain.
A function is neither if it meets neither relationship.
Use a rule or full graph to check the relationship throughout the domain.

Check your understanding

Question 1

A function has domain all real numbers, satisfies

h(-x)=h(x)

for every real

x

, and has

h(2)=3

. How is it classified?

Even
Odd
Neither
It cannot be classified without its graph

Show answer and explanation

Even

Its domain is symmetric about zero, and it satisfies the defining relationship for an even function. The nonzero value also rules out the zero function, which would satisfy both relationships.

Question 2

For

p(x)=x^3-2x

, what is the classification?

Even
Odd
Neither
The rule is undefined for negative inputs

Show answer and explanation

Odd

The domain is all real numbers. Substitution gives

p(-x)=-x^3+2x=-(x^3-2x)=-p(x)

, so the function is odd.

Question 3

A function is defined only for the input

1

. Which statement follows from the standard even and odd definitions?

It is even because there are no unequal outputs to compare
It is odd because there are no equal outputs to compare
It is neither because it has a nonzero input
Its domain is not symmetric about zero, so it is neither even nor odd

Show answer and explanation

Its domain is not symmetric about zero, so it is neither even nor odd

The domain contains

1

but not its opposite,

-1

. It is not symmetric about zero, so it does not meet the domain requirement for either standard classification.

Key terms

Domain: The set of input values for which a function is defined.
Opposite inputs: Inputs such as $x$ and $-x$ that have equal distance from zero and opposite signs.
Domain symmetric about zero: A domain that contains the opposite of each of its inputs.
Even function: A function with a domain symmetric about zero whose outputs satisfy $f(-x)=f(x)$ .
Odd function: A function with a domain symmetric about zero whose outputs satisfy $f(-x)=-f(x)$ .
Symmetry about the $y$ -axis: A graph pattern where reflecting across the $y$ -axis leaves the graph unchanged.
Symmetry about the origin: A graph pattern where turning the graph halfway around the origin leaves it unchanged.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C1.9. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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