A function is a mapping from one set - the domain - to another set - the codomain or range - such that each element in the domain is associated with one and only one element in the range.
The domain and codomain need not be different.
It is possible for several elements in the domain to be mapped onto the same element in the range ie a "many-to-one" mapping. However a "one-to-many" mapping not permitted. It may be possible to redefine the domain or range of a one-to-many mapping so that it is no longer is one-to-many and so becomes a function.
For example,
f(x) = x2 (for real x) is a perfectly legitimate many-to-one function. Both -2 and +2 are mapped to 4, but that is OK.
f(x) = sqrt(x) for x ≥ 0 is not a function because 4 can be mapped to -2 or +2. To avoid this, you can restrict the range to f(x) ≥ 0 or define f(x) = |sqrt(x)|.
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A range used as a math term means subtracting the highest value from the lowest value. the set of values that a given function can take as its argument varies.
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"Subset" IS the math term in this case.
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.
"key concept" is not a mathematical term . A key concept is an important idea. Two key concepts in math are derivatives and integrals of functions. ( "function" is also a key concept.)