Functions

To determine $g(x)$, let's first understand the given functions:We have $f(x)=\sqrt{x-1}$. This means $f(x)$ is a function that transforms $x$ into $\sqrt{x-1}$.Next, the composition $g(f(x))=x+2x^2+1$ can be rewritten using substitution and simplification steps:Begin by rewriting the expression for $g(f(x))$ :$g(f(x))=x+2\sqrt{x-1}+1$Recognize a perfect square identity in:$x+2\sqrt{x-1}+1=(\sqrt{x-1}+1)^2$Here, $(\sqrt{x-1}+1)^2=\sqrt{x-1}^2+2\sqrt{x-1}+1$, which simplifies to $x-1+2\sqrt{x-1}+1=x+2\sqrt{x-1}$. Notice then that:$g(f(x))=(\sqrt{x-1}+1)^2=((\sqrt{x-1}-1)+2)^2=(f(x)+2)^2$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 .