# 005 Sample Final A, Question 3

Question Find f ${\displaystyle \circ }$ g and its domain if ${\displaystyle f(x)=x^{2}+1\qquad g(x)={\sqrt {x-1}}}$

Foundations:
1) How do you compose two functions, such as given ${\displaystyle f=x^{2}}$  and   ${\displaystyle g=x+1}$, what is f${\displaystyle \circ }$g?
2) When should a point x be in the domain of f${\displaystyle \circ }$g?
1) We replace all occurrences of x in f with g, so ${\displaystyle f\circ g=(x+1)^{2}}$.
2) A point should be in the domain of f${\displaystyle \circ }$g when it is in the domain of g, and g(x) is in the domain of f.
First we find the domain of g. Since f ${\displaystyle \circ }$ g = f(g(x)). So if x is not in the domain of g, it is not in the domain of f ${\displaystyle \circ }$ g. The domain of g is ${\displaystyle [1,\infty )}$.
To find f ${\displaystyle \circ }$ g we replace any occurrence of x in f with g, to yield ${\displaystyle ({\sqrt {x-1}})^{2}+1=x-1+1=x}$
f ${\displaystyle \circ }$ g = ${\displaystyle x}$, and the domain is ${\displaystyle [1,\infty )}$.