How do I check if a function is even or odd?

Substitute -x for x throughout the function and simplify. If f(-x) = f(x), it's even. If f(-x) = -f(x), it's odd. If f(-x) doesn't match either of these, the function is neither.

Can a function be both even and odd?

Yes, but only the zero function f(x) = 0 is both even and odd. Every other function can be only one or the other, or neither.

Why does it matter if a function is odd?

Odd functions have 180° rotational symmetry around the origin, meaning you only need to graph one side (say, x > 0) and then rotate it to get the rest. In calculus, integrals of odd functions over symmetric intervals simplify, and odd functions always pass through the origin.

What does it mean to say a function is even?

An even function is symmetric about the y-axis: f(-x) = f(x) means the left half of the graph is a mirror image of the right half. This lets you sketch the full graph by mirroring half of it, and in calculus, integrals of even functions have special simplifications.

Even and Odd Functions

Determine whether a function is even ( $f(-x) = f(x)$ ) or odd ( $f(-x) = -f(x)$ ) to identify graph symmetry and simplify calculations.

f(-x) = f(x) \quad \text{or} \quad f(-x) = -f(x)

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What each symbol means

What Even and Odd Functions takes

Symbol	Meaning
$f$	A function that takes an input and produces an output; think of it as a rule or machine that processes numbers.
$x$	The input value to the function (the independent variable); it can be any real number in the domain, including negative values.

When to use it

Use this when you need to analyze function symmetry or quickly sketch a graph by exploiting mirror or rotational symmetry.

Level

Usually taught in: Algebra II

Worked examples

1. Determine if a simple quadratic is even or odd

Problem

Determine whether

f(x) = 5x^2 - 1

is even, odd, or neither.

$f(-x) = 5(-x)^2 - 1$
Substitute $-x$ for every instance of $x$ in the function.
$f(-x) = 5x^2 - 1$
Simplify using the exponent rule $(-x)^2 = x^2$ .
$f(-x) = f(x)$
The result $5x^2 - 1$ equals the original function.

Answer: $f \text{ is even}$

Since $f(-x) = f(x)$ , the function is even. Even functions are symmetric about the y-axis: the left and right halves are mirror images.

2. Determine if a cubic with fractions is even, odd, or neither

Problem

Determine whether

f(x) = \frac{2x^3}{3} - 4x

is even, odd, or neither.

$f(-x) = \frac{2(-x)^3}{3} - 4(-x)$
Substitute $-x$ for every instance of $x$ in the function.
$f(-x) = \frac{2 \cdot (-x^3)}{3} + 4x$
Simplify the exponent using $(-x)^3 = -x^3$ , and simplify $-4(-x) = 4x$ .
$f(-x) = -\frac{2x^3}{3} + 4x$
Rewrite: $\frac{2 \cdot (-x^3)}{3} = -\frac{2x^3}{3}$ .
$-f(x) = -\left( \frac{2x^3}{3} - 4x \right) = -\frac{2x^3}{3} + 4x$
Negate the entire original function by distributing the negative to each term.
$f(-x) = -f(x)$
Since $-\frac{2x^3}{3} + 4x = -\frac{2x^3}{3} + 4x$ , the condition $f(-x) = -f(x)$ is satisfied.

Answer: $f \text{ is odd}$

Since $f(-x) = -f(x)$ , the function is odd. Odd functions have point symmetry about the origin: if you rotate the graph 180° around the origin, it looks identical. This also means negative inputs produce outputs that are negatives of the corresponding positive inputs.

3. Determine symmetry for a construction crew productivity model

Problem

A construction supervisor models the cumulative productivity difference between two crews as

f(x) = 3x^2 - 12

, where

x

hours represents how many more hours Crew A worked than Crew B. Determine whether the function is even or odd, and explain what the symmetry tells you about the crews.

$f(-x) = 3(-x)^2 - 12$
To check symmetry, substitute $-x$ for $x$ : this represents the scenario where Crew B instead worked that many more hours than Crew A.
$f(-x) = 3x^2 - 12$
Simplify: $(-x)^2 = x^2$ .
$f(-x) = f(x)$
The simplified form matches the original function.

Answer: $f \text{ is even}$

Because $f(-x) = f(x)$ , the productivity difference function is even. This means the magnitude of productivity difference depends only on the absolute hour difference between crews, not which crew worked more hours. If Crew A works 5 hours more, the productivity difference has the same magnitude as if Crew B worked 5 hours more—though the sign would be opposite from each crew's perspective. This symmetry reveals that both crews are equally capable of creating productivity shifts.

Common mistakes

Where Even and Odd Functions usually goes wrong

The mistake	Why it is wrong	The fix
Testing one example value instead of using the algebraic definition: 'Since $f(2) = f(-2)$ , the function is even.'	A function must satisfy the even/odd condition for ALL values of $x$ , not just one lucky example; one example proves nothing.	Always use the algebraic definition: compute $f(-x)$ by substituting $-x$ into the full formula, then compare to $f(x)$ and $-f(x)$ .
Forgetting to apply the negative sign to all terms when computing $f(-x)$ : for $f(x) = x^3 + 2x$ , writing $f(-x) = -x^3 + 2x$ .	You must substitute $-x$ for every instance of $x$ , so $2(-x) = -2x$ , not $2x$ . The negative applies to the entire term.	Carefully substitute $-x$ everywhere: $f(-x) = (-x)^3 + 2(-x) = -x^3 - 2x$ . Then check if this equals $f(x)$ or $-f(x)$ .
Confusing the sign of outputs with the even/odd property: thinking $f(x) = x^2 + 3$ is odd because it has 'a positive constant' or $f(x) = -x^2$ is even because it has 'a negative sign.'	Even and odd describe symmetry relationships ( $f(-x) = f(x)$ or $f(-x) = -f(x)$ ), not whether the output values are positive, negative, or contain negative signs.	Ignore the sign or appearance of the formula; focus only on the relationship between $f(-x)$ and $f(x)$ or $-f(x)$ .

Tips and when to use something else

Use algebra, not guessing: always substitute $-x$ into the full function and simplify algebraically—do not try to memorize which functions are even or odd.
Remember the visual meaning: even functions like $f(x) = x^2$ are symmetric about the y-axis (fold the graph left-right), while odd functions like $f(x) = x^3$ are symmetric about the origin (rotate 180°).
Not all functions are even or odd—most are neither. If $f(-x)$ doesn't match $f(x)$ or $-f(x)$ , the function is neither even nor odd; this is completely normal.
For Composite Function analysis, use the rule: even ∘ even = even, odd ∘ odd = odd, even ∘ odd = even. This lets you determine if a combination like $g(f(x))$ is even or odd without full expansion.

Related formulas

Function NotationFunction notation y = f(x) is a way to write and evaluate functions, showing how input values transform into output values using a concise symbolic format.Domain and RangeDomain lists all valid inputs for a function; range lists all possible outputs—use this to understand what a function can do.Function TransformationsFunction transformations describe how shifting, stretching, and flipping graphs of functions create new related functions from a parent function.Piecewise FunctionsA piecewise function uses different formulas for different input ranges, letting you define relationships that change behavior at specific boundaries.Composite FunctionComposite functions apply one function to the output of another, letting you combine multiple transformations into a single operation.

Frequently asked questions

How do I check if a function is even or odd?: Substitute $-x$ for $x$ throughout the function and simplify. If $f(-x) = f(x)$ , it's even. If $f(-x) = -f(x)$ , it's odd. If $f(-x)$ doesn't match either of these, the function is neither.
Can a function be both even and odd?: Yes, but only the zero function $f(x) = 0$ is both even and odd. Every other function can be only one or the other, or neither.
Why does it matter if a function is odd?: Odd functions have 180° rotational symmetry around the origin, meaning you only need to graph one side (say, $x > 0$ ) and then rotate it to get the rest. In calculus, integrals of odd functions over symmetric intervals simplify, and odd functions always pass through the origin.
What does it mean to say a function is even?: An even function is symmetric about the y-axis: $f(-x) = f(x)$ means the left half of the graph is a mirror image of the right half. This lets you sketch the full graph by mirroring half of it, and in calculus, integrals of even functions have special simplifications.

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Reviewed 2026-09-18