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a function is a one-to-one and many-to one relation
A formula or graph are two ways to describe a math function. How a math function is described depends on the domain of the function or the complexity of the function.
That sounds really comlicated.
A piecewise defined function is a function which is defined symbolically using two or more formulas
It is the function for which all the elements of the range of the function corresponds to exactly one element of the domain.