Given the functions \( f(x) = x - 1 \) and \( g(x) = \frac{2x - 1}{x - 1} \), let's explore their properties:

### 1. **Domain**:
- **Function \( f(x) \)**:  
  \( f(x) = x - 1 \) is a linear function, so its domain is all real numbers, \( \mathbb{R} \).
  
- **Function \( g(x) \)**:  
  \( g(x) = \frac{2x - 1}{x - 1} \) is a rational function. To find the domain, we need to avoid division by zero.  
  - \( x - 1 = 0 \Rightarrow x = 1 \)
  - Therefore, the domain of \( g(x) \) is all real numbers except \( x = 1 \). So, the domain is \( \mathbb{R} \setminus \{1\} \).

### 2. **Simplification of \( g(x) \)**:
- Check if the rational function can be simplified:  
  \( g(x) = \frac{2x - 1}{x - 1} \) cannot be simplified further, since \( 2x - 1 \) and \( x - 1 \) do not share common factors.

### 3. **Behavior of \( g(x) \)**:
- At \( x = 1 \), the function \( g(x) \) is undefined due to division by zero.
- For values of \( x \neq 1 \), \( g(x) \) behaves as a rational function, with potential vertical asymptote at \( x = 1 \).

Do you have any questions or would you like further details on how these functions behave?

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Here are 5 follow-up questions for you:
1. What happens to the values of \( g(x) \) as \( x \) approaches 1 from the left or right?
2. Can you find the intersection points of \( f(x) \) and \( g(x) \) graphically?
3. What is the behavior of \( g(x) \) as \( x \) tends towards infinity?
4. How does the graph of \( f(x) \) compare with that of \( g(x) \)?
5. Can you express the inverse of the function \( f(x) \)?

**Tip**: Always check for points where a function may be undefined, especially when dealing with rational expressions.

Understand the properties of the functions f(x) = x - 1 and g(x) = (2x - 1) / (x - 1), including their domains, behavior near undefined points, and simplification steps. Suitable for students in Grades 9-12, covering key algebraic concepts.

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