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.