Even and Odd Functions

Quick Reference

Property	Even Function	Odd Function
Algebraic test	$f(-x)=f(x)$ for all $x$ in domain	$f(-x)=-f(x)$ for all $x$ in domain
Symmetry	About the $y$-axis	About the origin
Power function $x^n$	Even when $n$ is even	Odd when $n$ is odd
Simple examples	$x^2,x^4,|x|$	$x,x^3,{\displaystyle \frac{1}{x}}$

Even Functions

A function $y=f(x)$ defined on a symmetric interval $(-a,a)$ is called even if changing the sign of any $x$ in that interval leaves the function value unchanged:

$$f(-x)=f(x)\text{for every }x\text{ in }(-a,a).$$

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 $y$-axis and continue the same distance on the other side, we reach another point on the graph. Equivalently, folding the coordinate plane along the $y$-axis makes the left and right halves of the graph coincide exactly.

Examples of even functions are $f(x)=x^2$ and $g(x)={\displaystyle \frac{1}{x^2}}-1$.

Odd Functions

A function $y=f(x)$ defined on a symmetric interval $(-a,a)$ is called odd if changing the sign of any $x$ in that interval changes only the sign, not the absolute value, of the function:

$$f(-x)=-f(x)\text{for every }x\text{ in }(-a,a).$$

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 $(x,y)$ on the graph, the point $(-x,-y)$ also lies on the graph. Rotating the graph 180° about the origin produces the same curve.

Examples of odd functions are $f(x)=x^3$ and $g(x)={\displaystyle \frac{1}{x}}$.

Worked Example

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	$f(-x)+g(-x)=f(x)+g(x)$
odd $+$ odd	odd	$f(-x)+g(-x)=-f(x)-g(x)=-(f+g)(x)$
odd $+$ even	neither (in general)	no symmetry is preserved
even $\times$ even	even	$f(-x)\cdot g(-x)=f(x)\cdot g(x)$
odd $\times$ odd	even	$(-f(x))\cdot (-g(x))=f(x)\cdot g(x)$
odd $\times$ even	odd	$(-f(x))\cdot g(x)=-(f\cdot g)(x)$

Proofs of all six rules

In each proof below, $f$ and $g$ are functions with the stated parity, defined on a common symmetric domain. We verify the definition directly.

1. even + even = even
   Let $f(-x)=f(x)$ and $g(-x)=g(x)$. Set $h=f+g$. Then:
2. odd + odd = odd
   Let $f(-x)=-f(x)$ and $g(-x)=-g(x)$. Set $h=f+g$. Then:
3. odd + even = neither (in general)
   A general argument: suppose $h=f+g$ with $f$ odd and $g$ even. If $h$ were even, then $h(-x)=h(x)$ would give $-f(x)+g(x)=f(x)+g(x)$, forcing $f(x)=0$ for all $x$. If $h$ were odd, then $h(-x)=-h(x)$ would give $-f(x)+g(x)=-f(x)-g(x)$, forcing $g(x)=0$ for all $x$. So unless $f$ or $g$ is identically zero, $h$ is neither.
   Concrete example: Let $f(x)=x$ (odd) and $g(x)=x^2$ (even), so $h(x)=x+x^2$. $$h(-1)=-1+1=0,h(1)=1+1=2.$$ Since $h(-1)\ne h(1)$, $h$ is not even. Since $h(-1)\ne -h(1)=-2$, $h$ is not odd. $✓$
4. even × even = even
   Let $f(-x)=f(x)$ and $g(-x)=g(x)$. Then:
5. odd × odd = even
   Let $f(-x)=-f(x)$ and $g(-x)=-g(x)$. Then: The two minus signs cancel, making the product satisfy the even condition.
6. odd × even = odd
   Let $f(-x)=-f(x)$ (odd) and $g(-x)=g(x)$ (even). Then:

For quotients the same sign rules apply, since $f/g=f\cdot (1/g)$ and $1/g$ has the same parity as $g$. 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 $x^n$ ($n$ a positive integer): even when $n$ is even, odd when $n$ 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 $3x^4-5x^2+7$, is even.
• A polynomial with only odd-degree terms, such as $2x^5-x^3+4x$, is odd.
• A polynomial mixing even and odd degrees, such as $x^2+x$, is neither.

Rational functions $p(x)/q(x)$: 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 $f$.

Proof

$E$ is even: $O$ is odd: $𝑬\mathbf{+}𝑶\mathbf{=}𝒇$: Uniqueness. Suppose $f=E_1+O_1=E_2+O_2$ with $E_1,E_2$ even and $O_1,O_2$ odd. Then $$E_1-E_2=O_2-O_1.$$ 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 $h(x)=h(-x)$ and $h(x)=-h(-x)$, so $h(x)=-h(x)$, giving $h(x)=0$ for all $x$. Therefore $E_1=E_2$ and $O_1=O_2$. $◼$

Frequently Asked Questions

Can a function be both even and odd?

Yes, but only the zero function $f(x)=0$ satisfies both $f(-x)=f(x)$ and $f(-x)=-f(x)$ for all $x$. 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?

If $f$ and $g$ are both odd, then $f(-x)=-f(x)$ and $g(-x)=-g(x)$. Computing the product at $-x$: The two negatives cancel, so the product satisfies the even condition. A concrete example: $x\cdot x^3=x^4$, which is even.

Is $y=x^n$ even or odd?

For any positive integer $n$: if $n$ is even then $(-x{)}^n=x^n$, so $f$ is even; if $n$ is odd then $(-x{)}^n=-x^n$, so $f$ 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 $[-a,a]$ is always zero, since the positive and negative areas cancel exactly. For an even function, the integral over $[-a,a]$ equals twice the integral over $[0,a]$. 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 $x$ is in the domain, then $-x$ must also be in the domain. If the domain is $[0,\mathrm{\infty })$, for instance, the function cannot be classified as even or odd in the standard sense, because there is no $-x$ 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: $$f(x)=\underset{\text{even part}}{\underbrace{\frac{f(x)+f(-x)}{2}}}+\underset{\text{odd part}}{\underbrace{\frac{f(x)-f(-x)}{2}}}.$$ This decomposition is unique and is useful in Fourier analysis.