005 Sample Final A, Question 3

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

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [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 = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } , and the domain is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [1, \infty)} .