Even and Odd Function Calculator
This even and odd function calculator puts −x in for every x and tells you what comes back — even, odd, both, or neither. When the answer is neither, it hands you the one number that proves it rather than just the word.
Type 2x² − 3 and you get even. Type x² + x and you get neither, along with x = 1, where the whole thing falls apart.
What Even and Odd Functions Are
Two definitions, and everything else follows from them:
f(−x) = f(x) → even
f(−x) = −f(x) → odd
An even function gives the same answer for a number and its negative, so its graph is a mirror image across the vertical axis. An odd function gives the opposite answer, so its graph looks the same after being turned half a circle about the origin.
Both are claims about every x, not about one or two. That single fact decides how each of these questions has to be answered, as you will see below.
Put −x In: The Brackets Are the Exam
More marks are lost here than anywhere else in the topic, and it is not the mathematics that loses them.
f(x) = x² → f(−x) = (−x)² = x² — not −x²
Every x becomes (−x) with the brackets written in before anything is simplified. An even power swallows the minus sign; an odd power keeps it. Write the substitution down with the brackets in place and the rest of the question does itself:
f(x) = 2x³ − 4x → 2(−x)³ − 4(−x) = −2x³ + 4x
That is exactly −f(x), so the function is odd.
Two Tests: Even, Odd, or Neither
The two tests are not opposites of each other, and this catches people out. Failing the even test does not make a function odd. It has to pass the odd test on its own account, and most functions pass neither.
Compare against f(x) first, then against −f(x) — and −f(x) means flipping the sign of every term, not just the first one. Half-doing that step is the second commonest error here, and this page prints −f(x) in full so there is nothing to half-do.
Take x² + x. Substituting gives x² − x. That is not x² + x, so not even. And −f(x) is −x² − x, which it is not either. So: neither.
Proving "Neither" Takes One Number
Here is the part worth knowing, and it turns on a genuine asymmetry in the logic.
Even and odd are claims about every x, so no amount of testing can establish them — you need the algebra. But neither is a claim about one x, so one number finishes the job completely:
f(x) = x² + x, at x = 1: f(1) = 2, f(−1) = 0
0 is not 2, so the function is not even. And 0 is not −2 either, so it is not odd. Both tests have failed at a single point, and that settles it for good — one counterexample destroys a claim about every x.
Notice the direction carefully, because the reverse is not true. Finding that f(2) = f(−2) proves nothing at all about evenness; it says only that those two particular numbers happen to agree. Agreement at a point is worthless as proof; disagreement at a point is conclusive. This calculator gives you that number every time the answer is neither, because "neither" without it is an assertion rather than an answer.
When the Domain Decides It
Sometimes the algebra never gets a turn, and this is the case almost nothing explains properly.
Take f(x) = √x. Substitute and you get √(−x), which does not match — so most people write "neither" and move on. The real reason is different and better: √x only accepts numbers from 0 upwards. It has a value at x = 1 and no value at all at x = −1. So f(−x) does not exist anywhere f(x) does, and there is simply nothing to compare.
A function can only have a chance at symmetry if its domain is symmetric about zero. The same thing happens to ln x, and to a fraction whose gap sits on one side only. The answer is still neither — but knowing why is the difference between following a rule and understanding one, and this page says which of the two reasons applies.
The Only Function That Is Both
Exactly one function is even and odd at the same time:
f(x) = 0
It works because 0 = −0, so both conditions hold together. Nothing else manages it: any other constant is even but not odd, and any function with a value anywhere fails one test or the other. It is a small fact, but examiners like it precisely because it cannot be guessed.
Reading It Off the Powers
For a polynomial you can often see the answer before substituting. If every term has an even power it is even; if every term has an odd power it is odd; mix the two and it is neither.
A constant counts as an even power, since 3 = 3x⁰. That is why x² + 3 is even while x³ + 3 is neither — the constant is the odd one out in the second.
Two warnings about this shortcut. It works for polynomials only, so 1/x, √x and |x| are outside it — and of those, 1/x is odd and |x| is even, neither of which the rule would tell you. And it is a way of seeing the answer, not a way of writing it: substitute anyway, because the substitution is what earns the marks. If you go on to compose two functions, these properties carry through — an even function of an odd function is even.
Using the Even and Odd Function Calculator
Type the function of x on one line. Use ^ for powers, / for division, and brackets wherever you would write them. An f(x) = at the front is fine and gets ignored.
You get f(−x) worked out, both tests run separately, and either the number that proves "neither" or the two points that show the domain is lopsided. Nothing is rounded, and when the two expressions match exactly the page says so rather than leaning on arithmetic it does not need.
Even and Odd Function Calculator FAQ
How do you tell if a function is even or odd?
How do you prove a function is neither even nor odd?
Is any function both even and odd?
Why is the square root of x neither even nor odd?
Brackets first, both tests separately — and if the answer is neither, find the number that says so.
