A function is odd if plugging in the negative of any input flips the sign of the output. In mathematical terms, f(−x) = −f(x) for every x in the function’s domain. This single rule is what separates odd functions from even functions and from the many functions that are neither. Visually, it means the graph has rotational symmetry around the origin: if you spin it 180 degrees, it looks exactly the same.
The Defining Rule
The formal condition is straightforward. For every value of x, the function must satisfy f(−x) = −f(x). An equivalent way to write this is f(x) + f(−x) = 0, which says the outputs at x and −x always cancel each other out. Both forms say the same thing, so use whichever clicks for you.
This requirement has an important implication: the domain must be symmetric around zero. If x is in the domain, then −x must be too. A function defined only for positive numbers, for instance, can’t be odd because there’s no negative input to test against.
Another subtle consequence is that most odd functions pass through the origin. If zero is in the domain, then f(0) must equal −f(0), which forces f(0) = 0. So the graph of an odd function typically crosses right through (0, 0).
Common Odd Functions
The simplest odd function is f(x) = x, the identity function. Negate the input and you negate the output. From there, any power function with an odd exponent is odd: x³, x⁵, x⁷, and so on. You can verify this quickly since (−x)³ = −x³.
In trigonometry, sine is the classic odd function. Plugging in −x gives sin(−x) = −sin(x). Tangent is also odd. Cosine, by contrast, is even because cos(−x) = cos(x). This odd/even split between sine and cosine shows up constantly in math and physics, so it’s worth memorizing.
Other examples include cube root functions and higher odd-integer roots, the hyperbolic sine function, and the error function used in statistics.
How to Tell by Looking at the Graph
Even functions are symmetric about the y-axis, like a mirror reflection. Odd functions have a different kind of symmetry: rotational symmetry about the origin. If you rotate the entire graph 180 degrees around the point (0, 0), it lands exactly on top of itself. Think of the graph of y = x³. The curve rising to the right and the curve falling to the left are mirror images through the origin, not across any axis.
This visual test is useful when you’re given a graph but not a formula. Pick any point on the curve, find its mirror through the origin (flip both the x and y coordinates), and check whether that point is also on the curve. If it works for every point, the function is odd.
Combining Odd and Even Functions
When you add or subtract two odd functions, the result is still odd. The same holds for even functions: the sum of two even functions is even. But if you add an odd function to an even function, the result is generally neither odd nor even. This is worth remembering because many real-world functions are built by combining simpler pieces.
Multiplication follows different rules. Two odd functions multiplied together produce an even function, not an odd one. (Think of x · x = x², which is even.) Two even functions multiplied together stay even. The only way multiplication gives you an odd result is when you multiply an odd function by an even function.
For composition, plugging one odd function into another odd function gives an odd function. Plugging an odd function into an even function (or composing any function with an even function) gives an even function. Composing two odd functions preserves the odd symmetry because the sign flips twice, once in each function, keeping the overall relationship intact.
Odd Functions in Calculus
Odd symmetry has a powerful shortcut in integration. When you integrate an odd function over a symmetric interval, say from −a to a, the result is always zero. The positive area on one side of the origin exactly cancels the negative area on the other side. This can save significant work: if you recognize the function is odd and the limits are symmetric, you can write down zero without computing anything.
Derivatives also interact neatly with parity. The derivative of an odd function is an even function. You can see this intuitively: x³ is odd and its derivative, 3x², is even. Going in the other direction, the derivative of an even function is odd. Cosine is even, and its derivative, negative sine, is odd.
Odd Functions in Power Series
When you expand an odd function as a power series centered at zero, only odd powers of x appear. The series for sine, for example, is x − x³/6 + x⁵/120 − x⁷/5040 + … with no even-powered terms at all. Compare that to cosine, an even function, whose series contains only even powers: 1 − x²/2 + x⁴/24 − x⁶/720 + …. This pattern holds universally. If you’re building a power series and want the result to be odd, you need to leave out every even-powered term, including the constant.
This also explains the f(0) = 0 rule from another angle. A constant term in the series would be an x⁰ (even power) term, which would break the odd symmetry.
Functions That Are Neither Odd nor Even
Most functions don’t fall cleanly into either category. The exponential function eˣ, for instance, is neither odd nor even. However, any function defined on a symmetric domain can be split into an odd part and an even part. The even part is [f(x) + f(−x)]/2 and the odd part is [f(x) − f(−x)]/2. Added together, they reconstruct the original function. This decomposition is the basis for many techniques in signal processing and physics, where separating symmetric and antisymmetric components simplifies the math considerably.