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Let's analyze the given problem and solve it step by step.

### Given Functions:
- \( f(x) = \sqrt{x - 1} \)
- \( g(x) = -x \)

We are tasked with finding the domain of the following expressions using interval notation:
1. \( (f + g)(x) \)
2. \( (f - g)(x) \)
3. \( (f \cdot g)(x) \)
4. \( \left( \frac{f}{g} \right)(x) \)

### Step 1: Domain of Individual Functions
- **Domain of \( f(x) = \sqrt{x - 1} \)**: The square root function is only defined for non-negative arguments. Thus, we need \( x - 1 \geq 0 \), or \( x \geq 1 \).
  - **Domain of \( f(x) \)**: \( [1, \infty) \)

- **Domain of \( g(x) = -x \)**: Since \( g(x) \) is a linear function, it is defined for all real numbers.
  - **Domain of \( g(x) \)**: \( (-\infty, \infty) \)

The domain of any combination of functions depends on the intersection of their individual domains.

### Step 2: Domain of the Combined Functions

#### (a) \( (f + g)(x) \) and \( (f - g)(x) \)
- For \( f(x) + g(x) \) and \( f(x) - g(x) \), both functions are involved. Since the domain of \( f(x) \) is \( [1, \infty) \) and \( g(x) \) is defined everywhere, the domain of \( f + g \) and \( f - g \) will be limited by the stricter domain, which is \( [1, \infty) \).

- **Domain of \( (f + g)(x) \)** and **Domain of \( (f - g)(x) \)**: \( [1, \infty) \)

#### (b) \( (f \cdot g)(x) \)
- For the product \( f(x) \cdot g(x) \), we again have to consider the more restrictive domain. Since \( f(x) \) is only defined for \( x \geq 1 \), the domain of the product is still \( [1, \infty) \).

- **Domain of \( (f \cdot g)(x) \)**: \( [1, \infty) \)

#### (c) \( \left( \frac{f}{g} \right)(x) \)
- For the quotient \( \frac{f(x)}{g(x)} \), we must first ensure that \( g(x) \neq 0 \). Since \( g(x) = -x \), \( g(x) = 0 \) when \( x = 0 \). However, we also need to respect the domain of \( f(x) \), which requires \( x \geq 1 \). Therefore, the domain of \( \frac{f}{g} \) will be \( [1, \infty) \), since \( x = 0 \) is not within this interval.

- **Domain of \( \left( \frac{f}{g} \right)(x) \)**: \( [1, \infty) \)

### Final Results:
- **(a) \( (f + g)(x) \) Domain**: \( [1, \infty) \)
- **(b) \( (f - g)(x) \) Domain**: \( [1, \infty) \)
- **(c) \( (f \cdot g)(x) \) Domain**: \( [1, \infty) \)
- **(d) \( \left( \frac{f}{g} \right)(x) \) Domain**: \( [1, \infty) \)

Would you like further details or clarification on any of these points

Let f(x) = √(x - 1) and g(x) = -x. Find the domain for each domain using interval notation.

Learn how to find the domain of function combinations involving a square root function f(x) = √(x - 1) and a linear function g(x) = -x. This problem covers finding domains of function sums, differences, products, and quotients using interval notation, suitable for Grades 10-12.

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