#### Answer

The average rate of change of $f\left( x \right)=\sqrt{x}$ from ${{x}_{1}}=4\text{ to }{{x}_{2}}=9$ is $\frac{1}{5}$.

#### Work Step by Step

Consider the provided function $f\left( x \right)=\sqrt{x}$.
The value of function $f\left( x \right)=\sqrt{x}$ at ${{x}_{1}}=4$ is
$\begin{align}
& f\left( 4 \right)=\sqrt{\left( 4 \right)} \\
& =2
\end{align}$
The value of function $f\left( x \right)=\sqrt{x}$ at ${{x}_{2}}=9$ is
$\begin{align}
& f\left( 9 \right)=\sqrt{\left( 9 \right)} \\
& =3
\end{align}$
The average rate of change of $f$ from ${{x}_{1}}=4\text{ to }{{x}_{2}}=9$ is
$\begin{align}
& \frac{\Delta y}{\Delta x}=\frac{f\left( 9 \right)-f\left( 4 \right)}{9-4} \\
& =\frac{3-2}{9-4} \\
& =\frac{1}{5}
\end{align}$
Thus, the average rate of change of $f\left( x \right)=\sqrt{x}$ from ${{x}_{1}}=4\text{ to }{{x}_{2}}=9$ is $\frac{1}{5}$.