Formal Definition
Standard mathematical notation allows a formal definition of range.
In the first sense, the range of a function must be specified; it is often assumed to be the set of all real numbers, and {y | there exists an x in the domain of f such that y = f(x)} is called the image of f.
In the second sense, the range of a function f is {y | there exists an x in the domain of f such that y = f(x)}. In this case, the codomain of f must be specified, but is often assumed to be the set of all real numbers.
In both cases, image f ⊆ range f ⊆ codomain f, with at least one of the containments being equality.
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