Even And Odd Signals

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Even functions

ƒ(x) = x² is an example of an even function.

Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x in the domain of f:

Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.
Examples of even functions are |x|, x², x⁴, cos(x), and cosh(x).

 Odd functions

ƒ(x) = x³ is an example of an odd function.

Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x in the domain of f:

or

Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.
Examples of odd functions are x, x³, sin(x), sinh(x), and erf(x).

 Some facts

ƒ(x) = x³ + 1 is neither even nor odd.

A function's being odd or even does not imply differentiability, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist.

[edit] Basic properties

• The only function which is both even and odd is the constant function which is equal to zero (i.e., f(x) = 0 for all x).
• The sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over the given domain.
• The sum of two even functions is even, and any constant multiple of an even function is even.
• The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
• The product of two even functions is an even function.
• The product of two odd functions is an even function.
• The product of an even function and an odd function is an odd function.
• The quotient of two even functions is an even function.
• The quotient of two odd functions is an even function.
• The quotient of an even function and an odd function is an odd function.
• The derivative of an even function is odd.
• The derivative of an odd function is even.
• The composition of two even functions is even, and the composition of two odd functions is odd.
• The composition of an even function and an odd function is even.
• The composition of any function with an even function is even (but not vice versa).
• The integral of an odd function from −A to +A is zero (where A is finite, and the function has no vertical asymptotes between −A and A).
• The integral of an even function from −A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between −A and A).
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