Question

f(x) = x + 1, g(x) = √(x − 3) ; What is the domain of f ◦ g?
I just want to know if it's [3,infinity) or [2,infinity) because of the restrictions on compositions of functions.

Answer 1

Usually what i do is compose them , and don't simplify it. then look for the composed domain

Answer 2

f ◦ g(x) = f(g(x)) = f ( √(x − 3)) = √(x − 3) + 1

without simplify, the domain is x >= 3

Answer 3

the reason why i say don't simplify, let me illustrate with an example.
suppose you are given f(x) = x^2 , g(x) = √ (x-3)
then f(g(x)) = (√(x-3) )^2
if you simplify this you get x-3. The domain of x -3 is all the real numbers.
However that is not the domain of f ◦ g , the domain is x >= 3.

Answer 4

Perfect. The textbook is wrong-- just as I thought.

Thanks!