https://file.aiquickdraw.com/mathbot/file/1726216519888oc7wl3wc.jpg

Let's break down this problem step by step:

We are given two functions:
- \( f(x) = \sqrt{x - 1} \)
- \( g(x) = \sqrt{x - 2} \)

### 1. Domain of \( f(x) \) and \( g(x) \):

For the square root functions, the expressions under the square root must be non-negative.

- For \( f(x) = \sqrt{x - 1} \), we need \( x - 1 \geq 0 \), so \( x \geq 1 \). Therefore, the domain of \( f(x) \) is \( [1, \infty) \).
  
- For \( g(x) = \sqrt{x - 2} \), we need \( x - 2 \geq 0 \), so \( x \geq 2 \). Therefore, the domain of \( g(x) \) is \( [2, \infty) \).

### 2. Operations on \( f \) and \( g \):

We now find the domain for the following operations: \( f + g \), \( f - g \), \( f \cdot g \), and \( \frac{f}{g} \).

- **For \( f + g \) and \( f - g \):**
  
  The domain will be the intersection of the domains of \( f(x) \) and \( g(x) \), which is \( [2, \infty) \).

- **For \( f \cdot g \):**
  
  The domain is also the intersection of \( [1, \infty) \) and \( [2, \infty) \), which is \( [2, \infty) \).

- **For \( \frac{f}{g} \):**
  
  The domain is \( [2, \infty) \), but we must exclude any points where \( g(x) = 0 \) because division by zero is undefined. However, \( g(x) = \sqrt{x - 2} \) is only zero when \( x = 2 \). So, the domain of \( \frac{f}{g} \) is \( (2, \infty) \).

### Final answer:
- The domain of \( f + g \), \( f - g \), and \( f \cdot g \) is \( [2, \infty) \).
- The domain of \( \frac{f}{g} \) is \( (2, \infty) \).

It looks like the answer marked in the image (for \( \frac{f}{g} \)) is correct: \( (2, \infty) \).

Would you like more details or have any further questions?

### 5 Related Questions:
1. How do you find the domain of a square root function?
2. Why does \( g(x) = \sqrt{x - 2} \) impact the domain of \( \frac{f}{g} \)?
3. What happens if \( g(x) = 0 \) when dividing functions?
4. Can domains of different types of functions overlap in more complex cases?
5. How would the domain change if \( f(x) \) or \( g(x) \) included additional terms (like a polynomial)?

**Tip:** Always check for division by zero when determining the domain of rational functions!

Find f + g, f - g, fg, and f/g. Determine the domain for each function where f(x) = √(x - 1) and g(x) = √(x - 2).

Learn how to find the domains of combined functions f + g, f - g, fg, and f/g where f(x) = √(x - 1) and g(x) = √(x - 2). Understand key concepts such as the domain of square root functions, the intersection of domains, and handling division by zero.

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