How do I build a function of a given range and domain

Question

The question has been answered before but I do not understand it fully,

Give an example of a function, f(x), with a domain of (0,5] and a range of [0,∞)

I found the solution for it but my question is, does the value of the nominator matter at all when the X is at the denominator for example

sqr(5/x-1)

Answer 1

Well, if your answer is y=sqrt[5/(x-1)], you have a problem.

The domain of that function is x>1 and the range is (0,∞).

Try y =(x-1)^2* (sqrt(x*(5-x))/((x-2)^2)

What is the "nominator"? If you mean the numerator, it certainly does matter! Look at the answer I gave you.

The numerator defines the domain since the terms under the radical are defined only for (0,5]. The x-1 in te numerator is necessary to be sure 0 is in the range; without the x-1, the range is only the open interval (0,∞). The denominator is what gives you the unbounded range.