Function Odd/Even/Neither Checker

The Function Odd/Even/Neither Checker classifies a function f(x) as even, odd, both, or neither by testing it at five sample points and their negatives, then comparing f(x), f(-x), and -f(x) at each one.

It's built for algebra and precalculus students checking symmetry homework, and anyone who wants to confirm a function's symmetry without plugging in values by hand.

How to use it

1. Enter f(x) in terms of x — standard math notation works: ^ for powers, plus sqrt(), sin(), cos(), log(), and the other functions supported by mathjs's expression parser. The function must be defined for both positive and negative x.
2. Read Classification for the verdict — Even, Odd, Both, or Neither.
3. f(x) vs. f(-x) at Sample Points shows the comparison table behind that verdict.
4. Explanation spells out the reasoning in plain language.

Results update automatically as you type. For example, f(x) = x^2 returns Even (f(-x) = f(x) at every sample point), f(x) = x^3 returns Odd (f(-x) = -f(x) at every sample point), and f(x) = x^2 + x returns Neither.

What does even, odd, both, and neither mean?

• Even: f(-x) = f(x) for every x in the domain — the graph is a mirror image across the y-axis. Example: x^2, cos(x).
• Odd: f(-x) = -f(x) for every x in the domain — the graph is symmetric about the origin (rotating it 180° leaves it unchanged). Example: x^3, sin(x).
• Both: only the zero function, f(x) = 0, satisfies both conditions at once.
• Neither: most functions, including any that mix even and odd terms — x^2 + x is neither, since the x^2 term is even but the x term is odd.

How does the checker decide?

It evaluates f(x) at five fixed, non-integer sample points (0.7, 1.3, 2.1, 3.4, 4.6) and their negatives, then checks whether f(-x) matches f(x) at every point (even), matches -f(x) at every point (odd), both (the zero function), or neither. Non-integer points are used deliberately — a function that happens to agree at whole numbers by coincidence would otherwise read as symmetric when it isn't. This is a numerical check over sample points, not a symbolic proof: it's reliable for standard algebraic and trigonometric functions, but a pathological function that's symmetric at these five points and nowhere else would be misclassified. If your function isn't defined for negative x (like sqrt(x) or log(x)), it can't be classified — even/odd symmetry requires a domain symmetric about 0.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials (x^3 - 2x + 1), trigonometric functions (sin(x), cos(x), tan(x)), roots and logarithms (sqrt(x), log(x)), and combinations of these. Angles in trig functions are in radians by default.

To compose two functions together instead, see the Function Composition Calculator. To evaluate arbitrary math expressions directly, see the Math Evaluator.

Privacy

This calculator runs entirely in your browser. Your function is never uploaded, logged, or stored — the computation happens locally on your device.