Answer
The average rate of change of $f\left( x \right)=\sqrt{x}$ from ${{x}_{1}}=9\text{ to }{{x}_{2}}=16$ is $\frac{1}{7}$.
Work Step by Step
Consider the provided function: $f\left( x \right)=\sqrt{x}$.
The value of the function $f\left( x \right)=\sqrt{x}$ at ${{x}_{1}}=9$:
$\begin{align}
& f\left( 9 \right)=\sqrt{\left( 9 \right)} \\
& =3
\end{align}$
The value of the function $f\left( x \right)=\sqrt{x}$ at ${{x}_{2}}=16$:
$\begin{align}
& f\left( 16 \right)=\sqrt{\left( 16 \right)} \\
& =4
\end{align}$
The average rate of change of $f$ from ${{x}_{1}}=9\text{ to }{{x}_{2}}=16$ is:
$\begin{align}
& \frac{\Delta y}{\Delta x}=\frac{f\left( 16 \right)-f\left( 9 \right)}{16-9} \\
& =\frac{4-3}{7} \\
& =\frac{1}{7}
\end{align}$
Thus, the average rate of change of $f\left( x \right)=\sqrt{x}$ from ${{x}_{1}}=9\text{ to }{{x}_{2}}=16$ is $\frac{1}{7}$.
