Composite functions

1)When f(x) and g(x) are given as expressions you can just replace x in f by g(x) and it gives an expression for f(g(x)).

Let f(x) = x²  , g(x) =  √(x – 1)                                      f(g(x)) = x – 1 and f(g(0))= -1

2)When domains of f and g given you can find the range of g , and f(g(x)) exists if the range of g is a subset of domain of f  . In this case domain of f(g(x)) is same as the domain of the function g.

In the previous example if domain of f given as all real numbers and domain of g as real numbers not less than 1 ,  range of g is set of positive real numbers including 0 and it is subset of domain of f . Therefore f(g(x)) can be defined. But f(g(0)) can not be defined because 0 is not in the domain of the composite function. Since range of f is not a subset of domain of g ,  g(f(x)) can not be defined.

3)If you are asked to find f(g(x)) with it’s domain without giving domains of f and g , you need to restrict the domain of g in order to make range of g to be a subset of domain of f.

Let f(x) = √(x -1) , g(x) = x²                                       here you need to restrict domain of g to             (-∞ , 1] ∪  [ 1 , ∞ ) in order have range of g that is [ 1  ,  ∞ ) as a subset of domain of f.

Hope this answer would be sufficient to get the clear picture of the issue. I think better way to get this fully understood is to practice for few more problems.  Appreciate your suggestions regarding this answer. Because of a system error sometimes your reply can be taken as another answer.