A function is even when plugging in a negative input gives you the exact same output as the positive version. In algebraic terms, f(x) = f(−x) for every x in the function’s domain. This single rule is the complete definition, and everything else about even functions flows from it.
The Algebraic Definition
The formal test is straightforward: replace every x in the function with −x. If the result simplifies back to the original function, it’s even. Take f(x) = x². Substituting gives f(−x) = (−x)² = x², which matches the original. That makes x² an even function.
Now try f(x) = x³. Substituting gives f(−x) = (−x)³ = −x³, which is the negative of the original. That’s not even (it’s actually odd). And for something like f(x) = x² + x, substituting gives (−x)² + (−x) = x² − x, which matches neither f(x) nor −f(x). This function is neither even nor odd. Most functions fall into this “neither” category, so don’t assume every function has to be one or the other.
Y-Axis Symmetry
The geometric meaning of f(x) = f(−x) is that the graph is a mirror image across the y-axis. Pick any point on the right side of the graph. There will be a matching point at the same height on the left side, at equal distance from the y-axis. If you folded the graph along the vertical axis, the two halves would land exactly on top of each other.
This visual check is useful when you’re looking at a graph and want a quick read on whether a function might be even. If the curve looks symmetric across the y-axis, it probably is. But the algebraic test is the only way to confirm it with certainty.
The Exponent Shortcut for Polynomials
For polynomials, there’s a fast rule: if every term has an even exponent, the function is even. This covers x², x⁴, x⁶, and so on. Constants count too, because a constant like 5 is really 5x⁰, and zero is an even number.
So f(x) = 3x⁴ − 2x² + 7 is even. Every exponent (4, 2, and 0) is even. But f(x) = x⁴ + x is not, because that lone x term has an exponent of 1. A single odd-powered term is enough to break the symmetry. This shortcut only works cleanly for polynomials. For other types of functions, you need the full substitution test.
Common Even Functions
Beyond simple power functions, several important functions are even:
- Cosine: cos(x) = cos(−x) for all values of x. Its familiar wave is perfectly symmetric about the y-axis.
- Secant: Since secant is just 1/cos(x), and the reciprocal of an even function is even, secant inherits that property.
- Absolute value: |x| gives the same result whether you feed it 3 or −3.
- x to any even power: x², x⁴, x⁶, and so on.
For contrast, sine, tangent, cotangent, and cosecant are all odd functions, not even. Cosine is the only one of the basic trig functions that passes the even test.
How Even Functions Combine
Even functions stay even under most basic operations. Adding two even functions gives you an even function. Subtracting them does too. Multiplying or dividing two even functions also produces an even result. This means you can build complex even functions from simpler ones without losing the symmetry.
Composition has its own pattern. Plugging any function into an even function (as the outer function) always produces an even result. So if g(x) is even, then g(h(x)) is even regardless of what h looks like. The outer function controls the symmetry. Composing two even functions together also stays even.
Why It Matters in Calculus
Even functions have a property that can cut integration work in half. When you integrate an even function over a symmetric interval from −a to a, you can instead integrate from 0 to a and double the result:
∫ from −a to a of f(x) dx = 2 × ∫ from 0 to a of f(x) dx
This works because the area under the curve on the left side of the y-axis is identical to the area on the right. Rather than computing both halves, you compute one and multiply by two. On an exam or in applied work, recognizing an even function before integrating saves real time.
The Taylor series of an even function also reflects its symmetry. Only even powers of x appear in the expansion. Cosine, for example, expands to 1 − x²/2 + x⁴/24 − x⁶/720 + … with no x, x³, or x⁵ terms. Every odd-powered coefficient is zero, which is a direct consequence of the left-right symmetry built into the function.
How to Test Any Function
The process is the same every time. First, substitute −x for every instance of x in the function. Second, simplify completely. Third, compare the result to the original. If f(−x) equals f(x), the function is even. If f(−x) equals −f(x), it’s odd. If it matches neither, the function is neither even nor odd.
A few things to watch for: don’t forget to substitute −x into every occurrence of x, including inside exponents, denominators, and trig arguments. And remember that the test must hold for every x in the domain, not just a few convenient values. Checking one or two points can suggest evenness, but only the algebra proves it.