odd function
f(-x) = f(x), symmetric about the origin, sin, cos, sec
In mathematics, an odd function is a function that satisfies the property:
f(-x) = -f(x)
where x is any input value of the function f(x). In other words, the output of an odd function when evaluated at the opposite of any input value is equal to the opposite of the output obtained at that input value.
Geometrically, an odd function is one that is symmetric with respect to the origin, meaning that if the graph of the function is reflected across the origin, it is unchanged.
Examples of odd functions include:
– The sine function: sin(-x) = -sin(x)
– The tangent function: tan(-x) = -tan(x)
– The cube root function: cbrt(-x) = -cbrt(x)
One important property of odd functions is that their integral over any interval that is symmetric about the origin is always zero. In other words,
∫[-a,a] f(x) dx = 0
where a is any positive number. This property holds because the positive and negative areas of the function cancel each other out.
Lastly, any odd function can be decomposed into an odd part and an even part. The odd part is simply the original odd function, and the even part is equal to f(0) plus the even extension of f(x) (i.e., f(x) + f(-x))/2.
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