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In this problem, you are given a graph of the function \( y = f(x - 2) \) and asked to find the domain of the function:

\[
g(x) = \frac{1}{\sqrt{x f(x)}}
\]

Let's break this down step by step:

### 1. **Analyze the graph of \( f(x - 2) \):**
   - The function \( f(x - 2) \) represents a horizontal shift of the function \( f(x) \) to the right by 2 units.
   - The graph provided shows a curve that has a domain approximately from \( -4 \) to \( 3 \).

### 2. **Determine the domain of \( g(x) = \frac{1}{\sqrt{x f(x)}} \):**
   - For \( g(x) \) to be defined, the expression inside the square root, \( x f(x) \), must be **positive** (because the square root of a negative number is undefined, and division by zero is not allowed).
   - Therefore, \( x f(x) > 0 \), which means both \( x \) and \( f(x) \) must have the same sign (both positive or both negative).

### 3. **Identifying where \( x f(x) > 0 \):**
   - From the graph of \( f(x) \), we can visually determine where \( f(x) \) is positive or negative.
   - The important intervals are:
     - \( x < -1 \), where \( f(x) > 0 \).
     - \( -1 < x < 0 \), where \( f(x) < 0 \).
     - \( x > 0 \), where \( f(x) > 0 \).

### 4. **Choose the correct domain:**
   Based on the conditions for \( g(x) \) to be defined, the valid intervals for the domain are:
   \[
   (0, 1) \cup (2, 4)
   \]

Thus, the correct answer is option \( \boxed{2} \).

Would you like more details or have any questions? 

Here are some related questions:
1. How does a horizontal shift of a graph affect its domain?
2. What are the conditions for a function involving square roots to be defined?
3. How do you analyze the domain of a function given its graph?
4. What is the significance of the signs of \( x \) and \( f(x) \) in this context?
5. Can \( f(x) \) be zero in the domain of \( g(x) \)? Why or why not?

**Tip:** Always check the sign of the entire expression inside a square root and make sure there are no negative values.

If the graph of the function y = f(x - 2) is shown, what is the domain of the function g(x) = 1 / sqrt(x * f(x))?

Learn how to determine the domain of the function g(x) = 1 / sqrt(x * f(x)) when given the graph of y = f(x - 2). This problem involves function transformations and analyzing conditions for square root and division to be defined. Suitable for students in Grades 10-12.

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