Answer
See the explanation below.
Work Step by Step
(a)
Consider the function
$\begin{align}
& \left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right) \\
& =f\left( x+1 \right) \\
& =\sqrt{x+1}
\end{align}$
Therefore, the value of $\left( f\circ g \right)\left( x \right)$ for the functions $f\left( x \right)=\sqrt{x}$ and $g\left( x \right)=x+1$ is $\sqrt{x+1}$.
(b)
Consider the function
$\begin{align}
& \left( g\circ f \right)\left( x \right)=g\left( f\left( x \right) \right) \\
& =g\left( \sqrt{x} \right) \\
& =\sqrt{x}+1
\end{align}$
Therefore, the value of $\left( g\circ f \right)\left( x \right)$ for the functions $f\left( x \right)=\sqrt{x}$ and $g\left( x \right)=x+1$ is $\sqrt{x}+1$.
(c)
From part (a), $\left( f\circ g \right)\left( x \right)=\sqrt{x+1}$.
Therefore,
$\begin{align}
& \left( f\circ g \right)\left( 3 \right)=\sqrt{3+1} \\
& =\sqrt{4} \\
& =\pm 2
\end{align}$
Therefore, the value of $\left( f\circ g \right)\left( 3 \right)$ for the functions $f\left( x \right)=\sqrt{x}$ and $g\left( x \right)=x+1$ is $\pm 2$.
