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# What are piecewise defined functions?

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A piecewise-defined function consists of two or more subfunctions while maintaining the rule that a function has only y-value for each x-value. Some mathematicians call the piecewise function a hybrid function. Merchants use piecewise functions in describing pricing schemes that give discounts for volume sales.

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There are an infinite number of examples of piecewise functions, varying in complexity. The absolute value function is one of the simplest of these functions. Its formula is written f(x) = |x|. The subfunctions that define this function are f(x) = x if x > 0 and f(x) = -x if x < 0. The graph of this function is two rays, originating at the point (0,0), both at a 45-degree angle to the horizon.

The floor function forms an infinite number of steps and is a piecewise-defined function. In this function, f(x) is the integer greater than or equal to x.

Piecewise functions join functions of varying complexity, according to application. On one side of the gap, the function could be part of a parabola, while the other side is a straight line. In some piecewise functions, there are gaps between the two subfunctions where x is undefined, leaving a gap in the graph as well. If f(x) = 1/x, the function is undefined where x=0.

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