In mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another (the dependent variable), which changes along with it. Most functions are numerical; that is, a numerical input value is associated with a single numerical output value. The formula &math.A; = π&math.r;2, for example, assigns to each positive real number &math.r; the area &math.A; of a circle with a radius of that length. The symbols &math.f;(&math.x;) and &math.g;(&math.x;) are typically used for functions of the independent variable &math.x;.
A multivariable function such as &math.w; = &math.f;(&math.x;, &math.y;) is a rule for deriving a single numerical value from more than one input value. A periodic function repeats values over fixed intervals. If &math.f;(&math.x; + &math.k;) = &math.f;(&math.x;) for any value of &math.x;, &math.f; is a periodic function with a period of length &math.k; (a constant). The trigonometric functions are periodic. &Seealso; density function; exponential function; hyperbolic function; inverse function; transcendental function.
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