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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?

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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.

Begin by graphing the square root function, f(x) = √x. Then, use transformations of this graph to graph the given function g(x) = (1/3)√(x - 5). Choose the correct graph of g(x) from the options below.

Learn how to graph the transformed square root function g(x) = (1/3)√(x - 5) by applying horizontal shifts and vertical compression. This problem involves transformations of f(x) = √x, suitable for high school students in Grades 9-12.

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