odd function
f(-x)f(x) for all values of x in the domain of the function. The graph of the function will be symmetric about the origin.
An odd function is a type of function in mathematics that has a specific symmetry property. A function f(x) is said to be odd if for every value of x, the value of f(-x) is equal to the negative of f(x).
In mathematical terms, this can be expressed as:
f(-x) = -f(x)
In other words, if you reflect the graph of an odd function across the y-axis, then the resulting graph is indistinguishable from the original graph. For example, the function f(x) = -x is an odd function because:
f(-x) = -(-x) = x = -f(x)
Other examples of odd functions include:
– The sine function: sin(-x) = -sin(x)
– The tangent function: tan(-x) = -tan(x)
– The reciprocal of x: 1/-x = -1/x
Odd functions are interesting because they have some unique properties. For example, the integral of an odd function over a symmetric interval (such as [-a,a]) is always zero, since the positive and negative areas cancel out. Additionally, odd functions are often used in signal processing and other applications to filter out certain types of noise.
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