Even and Odd Functions

A function is even if its graph is symmetric about the y-axis, and odd if its graph is symmetric about the origin. Both properties are checked with a single algebraic test and have important consequences in calculus, physics, and engineering.

Quick Reference

Property	Even Function	Odd Function
Algebraic test	for all in domain	for all in domain
Symmetry	About the -axis	About the origin
Power function	Even when is even	Odd when is odd
Simple examples

Even Functions

A function is even if

A function defined on a symmetric interval is called even if changing the sign of any in that interval leaves the function value unchanged:

What does y-axis symmetry mean geometrically?

If a function is even, its graph is symmetric about the y-axis. That is, if we draw a horizontal segment from any point on the graph to the

-axis and continue the same distance on the other side, we reach another point on the graph. Equivalently, folding the coordinate plane along the

-axis makes the left and right halves of the graph coincide exactly.

Graph of an even function showing y-axis symmetry: the left and right halves mirror each other. — Graph of an even function is symmetric about the y-axis.

Examples of even functions are and .

Graph of f(x) = x squared, a parabola symmetric about the y-axis. — Graphs of some even functions.

Graph of g(x) = 1/x squared minus 1, symmetric about the y-axis. — Graphs of some even functions.

Odd Functions

A function is odd if

A function defined on a symmetric interval is called odd if changing the sign of any in that interval changes only the sign, not the absolute value, of the function:

What does origin symmetry mean geometrically?

If a function is odd, its graph is symmetric about the origin. That is, if we draw a segment from any point on the graph through the origin and continue the same distance on the other side, we reach another point on the graph. Equivalently, for any point

on the graph, the point

also lies on the graph. Rotating the graph 180° about the origin produces the same curve.

Graph of an odd function showing 180-degree rotational symmetry about the origin. — Graph of an odd function is symmetric about the origin.

Examples of odd functions are and .

Graph of f(x) = x cubed, an S-shaped curve through the origin. — Graphs of some odd functions.

Graph of g(x) = 1/x, a hyperbola symmetric about the origin. — Graphs of some odd functions.

Worked Example

Determine whether each function is even, odd, or neither:

Solution

(a) Compute

Since

, the function

is odd. Note that

has only odd-degree terms (

and

), which confirms the result.

Graph of y = x cubed minus 3x, an S-shaped curve with 180-degree rotational symmetry about the origin. — Graph of : symmetric about the origin (odd).

(b) Compute

Since

, the function

is even. Note that

has only even-degree terms (

and

), which confirms the result.

Graph of y = x to the fourth plus 2x squared, a wide U-shaped curve symmetric about the y-axis. — Graph of : symmetric about the -axis (even).

(c) Compute

Compare:

and

. Since

equals neither

nor

, the function

is neither even nor odd. This makes sense because

mixes an even constant (

) with an odd function (

), and by the odd+even rule the sum is neither.

Graph of h(x) = 1 minus 1 over x, showing no y-axis or origin symmetry. — Graph of : neither even nor odd.

Algebraic Properties: Sums and Products

When two functions with known parity are combined, the parity of the result follows fixed rules. These rules hold for all algebraic functions (polynomials, rational functions, power functions) and are proved directly from the definitions.

Sums and Products Table

Operation	Result	Why
even even	even
odd odd	odd
odd even	neither (in general)	no symmetry is preserved
even even	even
odd odd	even
odd even	odd

Proofs of all six rules

In each proof below,

and

are functions with the stated parity, defined on a common symmetric domain. We verify the definition directly.

even + even = even
Let and . Set . Then:
odd + odd = odd
Let and . Set . Then:
odd + even = neither (in general)
A general argument: suppose with odd and even. If were even, then would give , forcing for all . If were odd, then would give , forcing for all . So unless or is identically zero, is neither.
Concrete example: Let (odd) and (even), so . Since , is not even. Since , is not odd.
even × even = even
Let and . Then:
odd × odd = even
Let and . Then: The two minus signs cancel, making the product satisfy the even condition.
odd × even = odd
Let (odd) and (even). Then:

Key surprise: The product of two odd functions is even, not odd. The two negatives cancel. For example, (even), and (even). Think of it like multiplying signs: odd odd behaves like .

For quotients the same sign rules apply, since and has the same parity as . So odd/even is odd, even/odd is odd, even/even is even, and odd/odd is even.

Which Algebraic Functions Are Even or Odd?

Power functions ( a positive integer): even when is even, odd when is odd. This is the origin of the terminology.

Polynomials follow from the addition rules above:

A polynomial with only even-degree terms, such as , is even.
A polynomial with only odd-degree terms, such as , is odd.
A polynomial mixing even and odd degrees, such as , is neither.

Rational functions : combine the parities of numerator and denominator using the product/quotient rules in the table above.

Decomposing Any Function Into Even and Odd Parts

Every function defined on a symmetric domain can be written uniquely as the sum of an even function and an odd function. This is called the even–odd decomposition (or parity decomposition) of .

Theorem (Even–Odd Decomposition). Let be any function whose domain is symmetric about (i.e., ). Define

Then is even, is odd, and

Moreover, this decomposition is unique: there is only one way to write as a sum of an even function and an odd function.

Proof

is even:

is odd:

Uniqueness. Suppose

with

even and

odd. Then

The left side is even (difference of even functions) and the right side is odd (difference of odd functions). A function that is both even and odd satisfies

and

, so

, giving

for all

. Therefore

and

Find the even and odd parts of .

Solution

First compute

Now apply the formulas:

Check:

. ✓ Notice that

contains only even-degree terms, and

contains only odd-degree terms — exactly as the rules in the table above predict.

Frequently Asked Questions

Can a function be both even and odd?

Yes, but only the zero function

satisfies both

and

for all

. No nonzero function is both even and odd on a domain that contains both positive and negative values.

Why is odd times odd equal to even?

and

are both odd, then

and

. Computing the product at

The two negatives cancel, so the product satisfies the even condition. A concrete example:

, which is even.

even or odd?

For any positive integer

: if

is even then

, so

is even; if

is odd then

, so

is odd. This is precisely why functions with these symmetry properties were named "even" and "odd."

How does knowing a function is even or odd help in calculus?

In integration, the integral of an odd function over any symmetric interval

is always zero, since the positive and negative areas cancel exactly. For an even function, the integral over

equals twice the integral over

. These shortcuts are used constantly in physics, Fourier analysis, and signal processing, often cutting computation in half.

What if a function's domain is not symmetric about 0?

Even and odd symmetry requires the domain to be symmetric: if

is in the domain, then

must also be in the domain. If the domain is

, for instance, the function cannot be classified as even or odd in the standard sense, because there is no

counterpart to test against.

If I add an even and an odd function, what do I get?

In general, odd

even is neither even nor odd. However, every function defined on a symmetric domain can be decomposed into a sum of one even part and one odd part:

This decomposition is unique and is useful in Fourier analysis.

Even and Odd Functions

Even and Odd Functions

Quick Reference

Even Functions

Odd Functions

Worked Example

Algebraic Properties: Sums and Products

Sums and Products Table

Which Algebraic Functions Are Even or Odd?

Decomposing Any Function Into Even and Odd Parts

Frequently Asked Questions

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