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What is composite function in calculus?
composite of a function is fog(x)=f(g(x))
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.
How do you get a function of a function?
A function is a mapping from one set of numbers (domain) to another (range). The mapping need not be linear: it can be any mathematical function. That is, for every number in the domain the function provides a rule which allows you to calculate another number.If, then, you devise another function which is a mapping from the range of the first function to some other set, you have a function of a function.For example, suppose the first function, f, is "add 1" and the second function, g, is "square the number."Then the functiong of f = g[f(x)] = g[x+1] = [x+1]2 = x2 + 2x + 1however, note thatf of g = f[g(x)] = f[x2] = x2 + 1This illustrates that f of g is not the same as g of f.
Why can't the range of a function ever be equal to zero?
It most certainly can. In fact it can be quite a useful function. If you want to suppress one function, f(x), over part of its domain you could define another function, g(x) that is equal to zero over that part of the domain and then study the function: h(x) = f(x)*g(x) where both are defined = f(x) otherwise. You may want to do this if f(x) is ill-behaved over a part of its domain.
Does f have an inverse?
It very much depends on f. If f is one-to-one and onto (injective and surjective) then yes, else no. One-to-one means that for each element in the domain there is a different image in the range. This is not true for g(x) = x2 for example, where -3 and +3 are both mapped to +9. So g(x) does not have an inverse UNLESS you restrict the domain of g to non-negative reals. Then -3 is no longer in the domain. Onto means that every element in the range of the function has a corresponding element in the domain which is mapped onto it. Again, a suitable changes to the domain and range can transform a function without an inverse into an invertible one.
