Subject: Compostion of functions

Name: Yvonne

Who is asking: Teacher
Level: Secondary

Question: In our new text book, the following question occurs: State the domain and range of g(f(x))given that f(x) = -x² - 4 and g(x) = sqrt(x)

The range of f(x), x<=-4, is the domain of g(x). BUT, there is no solution in the Real numbers for g(f(x))= sqrt(-x² - 4).

In the solutions it says that this is not a function and therefore does not have a domain or range. Is it a relation? Is it anything?

The way I would put it is: The domain is the empty set (and so is the range).

So g(f(x)) is just the empty function, with empty domain and empty range. I guess that your textbook does not want to call that a "function"; but that is just a matter of taste, and it is not very important.