How would I go about doing this question?
Consider the functions f: [0, infinity) → R, f(x) = x-c and g: [-2, infinity) → R, g(x)=x^2 + c, where c is a real number. How do we find all the possible values for c for which f composition g and g composition f exist?
Quick question, when we're given this for example, f: [2, infinity) → R , what does this mean exactly? And how would we read this?
Thanks! 
I'll answer the second question before I get to the first - "f: X -->Y" can be read as "the function f where X is mapped onto Y". X is the domain, which you are hopefully familiar with. Y is the co-domain, which is what the range can exist in. I wouldn't worry too much about this, as methods hasn't ever seemed to give a question that requires you to have to compare the range to the co-domain. (feel free to correct me if anyone has seen a SAC that says otherwise)
Now, for the second question, let's first consider f(g(x)). For this function to exist,
. The range of g is
)
and the domain of f is
)
. By comparing the two, we see that out initial statement is satisfied when
- therefore, f(g(x)) exists if
)
.
Now, let's consider g(f(x)). For this functino to exist,
. Now, the range of f is going to be
)
, and the domain of g is
)
. Once again, by directly comparing the two, we see that the initial statement qualifies when
. So, g(f(x)) exists if
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