To determine g(x), let's first understand the given functions: We have f(x)=√x−1. This means f(x) is a function that transforms x into √x−1. Next, the composition g(f(x))=x+2x2+1 can be rewritten using substitution and simplification steps: Begin by rewriting the expression for g(f(x)) : g(f(x))=x+2√x−1+1 Recognize a perfect square identity in: x+2√x−1+1=(√x−1+1)2 Here, (√x−1+1)2=√x−12+2√x−1+1, which simplifies to x−1+2√x−1+1=x+2√x−1. Notice then that: g(f(x))=(√x−1+1)2=((√x−1−1)+2)2=(f(x)+2)2 Thus, we can deduce that g(x) must match the transformed output when x=f(x) plus an added constant: g(x)=(x+2)2 Therefore, g(x) is the function that squares the input, transformed by adding 2 .