Odd Functions
f(-x) = -f(x)
Odd functions are a type of function in mathematics that have a unique property. These functions are symmetric about the origin of the coordinate plane, meaning that for any point (x, y) on the graph of the function, the point (-x, -y) is also on the graph. This property can be expressed mathematically as:
f(-x) = -f(x)
where f(-x) represents the value of the function at -x, and -f(x) represents the negative of the value of the function at x.
Examples of odd functions include:
1. f(x) = x^3 – This is a simple example of an odd function. The graph of this function is a curve that passes through the origin of the coordinate plane and extends to the left and right in a symmetrical fashion.
2. g(x) = sin(x) – The sine function is also an odd function. Its graph is a curve that oscillates above and below the x-axis, crossing the origin in a symmetrical fashion.
3. h(x) = |x| – This is an example of a piecewise function that is odd only for values of x less than 0. For x greater than 0, the function is even.
One important property of odd functions is that their integral over a symmetric interval that includes the origin is always zero. This can be explained by the fact that the positive and negative areas on either side of the origin cancel each other out.
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