The question asks you to begin by graphing the square root function $f(x) = \sqrt{x}$, and then apply transformations to this graph to obtain the graph of the function $g(x) = \frac{1}{3} \sqrt{x - 5}$.

Step-by-Step Explanation of the Transformations:

1. Horizontal Shift: The function $g(x) = \frac{1}{3} \sqrt{x - 5}$ involves $(x - 5)$, which indicates a horizontal shift. Specifically, the graph of $f(x) = \sqrt{x}$ is shifted 5 units to the right.

2. Vertical Compression: The coefficient $\frac{1}{3}$ applied to the square root function indicates a vertical compression. The graph is compressed by a factor of $\frac{1}{3}$, meaning the $y$-values of all points on the graph are multiplied by $\frac{1}{3}$.

Matching the Transformation with the Given Options:

• The correct graph should show a square root curve that starts at $x = 5$ (due to the shift to the right by 5 units).
• The curve should be vertically compressed (less steep than the original $f(x) = \sqrt{x}$).

Looking at the graphs:

• Option C shows a square root graph that starts at $x = 5$ and is vertically compressed, matching the description of $g(x) = \frac{1}{3} \sqrt{x - 5}$.

Therefore, the correct answer is Option C.

Would you like more details on transformations or how to work through similar problems?

Here are five related questions to consider:

1. How do you determine horizontal shifts from a function's equation?
2. What is the effect of a coefficient less than 1 in front of a square root function?
3. Can vertical stretches and compressions also affect horizontal shifts?
4. How would the graph change if the equation were $g(x) = \frac{1}{3} \sqrt{x + 5}$?
5. What happens to the domain of the function when the graph is shifted horizontally?

Tip: When graphing functions, remember to adjust both the domain and the range according to any horizontal and vertical transformations.