Odd Even Function Checker
Odd · Even · Neither · Step-by-Step Proof · Symmetry Graph
Determine if f(x) is odd, even, or neither. Get a step-by-step algebraic proof, numerical verification table, and a symmetry graph — all instantly.
Quick Examples
Supports: +, -, *, /, ^, sin, cos, tan, abs, sqrt, x^n
Result
f(x)
f(-x)
Step-by-Step Proof
Numerical Verification Table
Evaluated at selected test points. A check mark means the condition holds at that x value.
| x | f(x) | f(-x) | -f(x) | f(-x) = f(x)? | f(-x) = -f(x)? |
|---|
Symmetry Graph
Enter up to 5 functions
Batch Summary
| # | f(x) | f(1) | f(-1) | Parity | Reason |
|---|
Common Functions Parity Reference
| Function | Parity | Rule / Reason |
|---|---|---|
| x² | Even | (-x)² = x² |
| x⁴, x⁶, x^(2n) | Even | Even-degree monomial |
| cos(x) | Even | cos(-x) = cos(x) |
| |x| | Even | |-x| = |x| |
| cosh(x) | Even | cosh(-x) = cosh(x) |
| x | Odd | (-x) = -x |
| x³, x⁵, x^(2n+1) | Odd | Odd-degree monomial |
| sin(x) | Odd | sin(-x) = -sin(x) |
| tan(x) | Odd | tan(-x) = -tan(x) |
| sinh(x) | Odd | sinh(-x) = -sinh(x) |
| x·sin(x) | Even | odd × odd = even |
| x·cos(x) | Odd | odd × even = odd |
| x² + x | Neither | Even + odd term |
| x + 1 | Neither | f(0)=1 ≠ 0 |
| e^x | Neither | e^(-x) ≠ e^x and ≠ -e^x |
| 0 (zero function) | Both | Only function that is both |
Parity Arithmetic Properties
Products
- Even × Even = Even
- Odd × Odd = Even
- Even × Odd = Odd
Sums
- Even + Even = Even
- Odd + Odd = Odd
- Even + Odd = Neither
Composition
- Even(Even) = Even
- Even(Odd) = Even
- Odd(Odd) = Odd
Derivatives & Integrals
- d/dx(Even) = Odd
- d/dx(Odd) = Even
- ∫₁₋a Even dx = 2∫₀ᵃ f dx
- ∫₁₋a Odd dx = 0
What Are Odd and Even Functions?
In mathematics, function parity refers to whether a function exhibits symmetry about the y-axis (even), symmetry about the origin (odd), or neither. These properties are among the most fundamental in analysis, applied mathematics, and physics.
A function f(x) is even if replacing x with -x returns the identical expression: f(-x) = f(x) for every x in the domain. Geometrically, this means the graph is a perfect mirror image about the y-axis — the right side reflects onto the left.
A function f(x) is odd if replacing x with -x negates the function: f(-x) = -f(x) for every x in the domain. Geometrically, the graph has 180-degree rotational symmetry about the origin — rotating the graph half a turn about (0, 0) leaves it unchanged.
Algebraic Test for Parity
The test is straightforward: substitute -x everywhere x appears, then simplify. Compare the simplified result to f(x) and -f(x):
- If the simplified f(-x) equals f(x) identically: even function
- If the simplified f(-x) equals -f(x) identically: odd function
- If neither match: neither
Example — testing f(x) = x³: Replace x with -x to get f(-x) = (-x)³ = -x³ = -f(x). Since f(-x) = -f(x), the function is odd.
Example — testing f(x) = x²: f(-x) = (-x)² = x² = f(x). Since f(-x) = f(x), the function is even.
Example — testing f(x) = x² + x: f(-x) = (-x)² + (-x) = x² - x. This equals neither f(x) = x² + x nor -f(x) = -x² - x. So the function is neither.
Geometric Meaning of Function Symmetry
The geometric interpretation is often the most intuitive:
- Even functions look the same on both sides of the y-axis. Think of a parabola x² or the cosine wave.
- Odd functions pass through the origin (f(0) = 0 is a necessary condition) and if you spin the graph 180° about the origin it lands back on itself. Think of x³ or the sine wave.
- Neither functions have no such symmetry. For example, f(x) = x + 1 is a line that does not pass through the origin and does not reflect onto itself.
Why Odd Functions Must Satisfy f(0) = 0
If f is odd and 0 is in its domain, setting x = 0 in f(-x) = -f(x) gives f(0) = -f(0), so 2f(0) = 0, meaning f(0) = 0. This is a quick sanity check: if you evaluate a candidate function at x = 0 and get a non-zero result, it cannot be odd.
Even-Odd Decomposition
A remarkable fact: every function can be uniquely decomposed into an even part and an odd part:
- Even part: E(x) = [f(x) + f(-x)] / 2
- Odd part: O(x) = [f(x) - f(-x)] / 2
So f(x) = E(x) + O(x). For example, e^x = cosh(x) + sinh(x) — the even part is the hyperbolic cosine and the odd part is the hyperbolic sine. This decomposition underlies Fourier series, where any periodic function is decomposed into cosines (even) and sines (odd).
Parity in Fourier Analysis and Integration
Function parity is invaluable in integration. For any symmetric interval [-a, a]:
- If f is even: the integral from -a to a equals 2 times the integral from 0 to a, halving computation.
- If f is odd: the integral from -a to a equals exactly 0, because positive and negative areas cancel perfectly.
In Fourier series, a function defined on [-L, L] can be represented as a sum of sines and cosines. If the function is even, only cosine terms appear (the Fourier cosine series). If the function is odd, only sine terms appear (the Fourier sine series). Identifying parity first dramatically reduces the work.
In physics, quantum mechanical wavefunctions in symmetric potentials (like the harmonic oscillator) are either even (symmetric states) or odd (antisymmetric states). Even parity wavefunctions have non-zero probability at the center, while odd parity wavefunctions have a node there.
Parity of Polynomial Functions
For polynomials, the parity rule is easy to remember: a polynomial is even if it contains only even-degree terms (constants count as degree 0, which is even). A polynomial is odd if it contains only odd-degree terms. Any polynomial mixing both even and odd degree terms is neither.
| Polynomial | Terms | Parity |
|---|---|---|
| 3x⁴ - 2x² + 5 | All even-degree (4, 2, 0) | Even |
| 7x⁵ - 4x³ + x | All odd-degree (5, 3, 1) | Odd |
| x³ + x² | Mixed (3 and 2) | Neither |
| x⁴ + 3 | All even-degree (4, 0) | Even |
| 2x + 1 | Mixed (1 and 0) | Neither |