005 Sample Final A, Question 3

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

Foundations:
1) How do you compose two functions, such as given $f=x^{2}$  and   $g=x+1$, what is f$\circ$g?
2) When should a point x be in the domain of f$\circ$g?
Answers:
1) We replace all occurrences of x in f with g, so $f\circ g=(x+1)^{2}$.
2) A point should be in the domain of f$\circ$g when it is in the domain of g, and g(x) is in the domain of f.

Step 1:
First we find the domain of g. Since f $\circ$ g = f(g(x)). So if x is not in the domain of g, it is not in the domain of f $\circ$ g. The domain of g is $[1,\infty )$.

Step 2:
To find f $\circ$ g we replace any occurrence of x in f with g, to yield $({\sqrt {x-1}})^{2}+1=x-1+1=x$

Final Answers:
f $\circ$ g = $x$, and the domain is $[1,\infty )$.