005 Sample Final A, Question 3

Question Find f $\circ$ g and its domain if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=x^{2}+1\qquad g(x)={\sqrt {x-1}}}

Foundations:
1) How do you compose two functions, such as given Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f=x^{2}}   and   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\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)} .