Odd and even functions are special types of functions that exhibit particular symmetry properties.
An even function is symmetric about the y-axis. Mathematically, a function $f(x)$ is even if:
$$f(-x) = f(x)$$
for all $x$ in the domain of $f$.
The function $f(x) = x^2$ is an even function because $f(-x) = (-x)^2 = x^2 = f(x)$.
Graphically, even functions can be recognized by their mirror symmetry about the y-axis.
An odd function has rotational symmetry of 180° about the origin. Mathematically, a function $f(x)$ is odd if:
$$f(-x) = -f(x)$$
for all $x$ in the domain of $f$.
The function $f(x) = x^3$ is an odd function because $f(-x) = (-x)^3 = -x^3 = -f(x)$.
Graphically, odd functions can be recognised by their rotational symmetry about the origin.
Not all functions are either odd or even. For example, $f(x) = x^2 + x$ is neither odd nor even.
Periodic functions can also be classified as odd or even. A function $f(x)$ is periodic with period $T$ if $f(x+T) = f(x)$.
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