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What is a function in math term?
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)|.
Is it ever possible for the domain and range to have different numbers of entries what happens when this is the case?
Yes. Typical example: y = x2. To avoid comparing infinite sets, restrict the function to integers between -3 and +3. Domain = -3, -2 , ... , 2 , 3. So |Domain| = 7 Range = 0, 1, 4, 9 so |Range| = 4 You have a function that is many-to-one. One consequence is that, without redefining its domain, the function cannot have an inverse.
Is it ever possible for the domain and range in a function to have different numbers of entries for example 3 domain entries to 5 range entries or 2 range entries to 7 domain entries?
Yes. The range can have fewer number of entries.As an extreme case, consider f(x) = 3, where x is a Real number.The domain is all Real numbers - infinitely many of them, while the range is one value: 3.A function can contain one-to-one or many-to-one relationships but one-to-many relationships are not permitted. As a result, the cardinality of the range cannot be bigger than the cardinality of the domain.
If F(x) x plus 2 and G(y) what is the domain of G(F(x))?
It is necessary to know the domain of x and also what the function G(y) is before it is possible to answer the question.
What is the domain and range of y equals cubed square root of x?
The simplest answer is that the domain is all non-negative real numbers and the range is the same. However, it is possible to define the domain as all real numbers and the range as the complex numbers. Or both of them as the set of complex numbers. Or the domain as perfect squares and the range as non-negative perfect cubes. Or domain = {4, pi} and range = {8, pi3/2} Essentially, you can define the domain as you like and the definition of the range will follow or, conversely, define the range and the domain definition will follow,
