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The problem asks to analyze the function \( f(x) = \frac{2x+1}{x-1} \). The task is to determine if the function is even, odd, or neither, and then to sketch its graph.

### 1. **Determine if the function is even, odd, or neither:**

A function is:
- **Even** if \( f(-x) = f(x) \) for all \(x\) in the domain.
- **Odd** if \( f(-x) = -f(x) \) for all \(x\) in the domain.

Let’s evaluate \( f(-x) \):

\[
f(-x) = \frac{2(-x) + 1}{-x - 1} = \frac{-2x + 1}{-x - 1}
\]

Now, compare \( f(-x) \) with \( f(x) \):

\[
f(x) = \frac{2x + 1}{x - 1}
\]

Clearly, \( f(-x) \neq f(x) \), so the function is **not even**.

Now, check if the function is odd:

For odd functions, we need \( f(-x) = -f(x) \). Let's calculate \( -f(x) \):

\[
-f(x) = -\left( \frac{2x + 1}{x - 1} \right) = \frac{-(2x + 1)}{x - 1} = \frac{-2x - 1}{x - 1}
\]

Compare this with \( f(-x) = \frac{-2x + 1}{-x - 1} \):

Since \( f(-x) \neq -f(x) \), the function is **not odd** either.

Thus, the function is **neither even nor odd**.

### 2. **Sketch the graph:**

The function \( f(x) = \frac{2x + 1}{x - 1} \) has a vertical asymptote where the denominator is zero, i.e., at \( x = 1 \).

#### Vertical asymptote:
- At \( x = 1 \), the function has a vertical asymptote because the denominator becomes zero and the function approaches infinity.

#### Horizontal asymptote:
- To find the horizontal asymptote, consider the limit as \( x \to \infty \) or \( x \to -\infty \). The degrees of the numerator and denominator are the same, so the horizontal asymptote is given by the ratio of the leading coefficients:
  
\[
\lim_{x \to \infty} f(x) = \frac{2}{1} = 2
\]

Thus, the horizontal asymptote is \( y = 2 \).

### Summary of features for the sketch:
- **Vertical asymptote** at \( x = 1 \).
- **Horizontal asymptote** at \( y = 2 \).
- The function is neither even nor odd.

Would you like more details on any part of this solution, or assistance with the sketch?

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Here are five related questions to deepen your understanding:
1. How can you find the domain of a rational function like \( f(x) \)?
2. What happens to the graph of a rational function near a vertical asymptote?
3. Can a rational function have more than one vertical asymptote?
4. How do you calculate slant asymptotes if they exist for rational functions?
5. What is the importance of horizontal asymptotes in determining end behavior of functions?

**Tip:** When checking if a function is even or odd, always consider the entire domain of the function and how it behaves under negation.

Diberikan fungsi f(x) = (2x + 1) / (x - 1). Apakah fungsi tersebut merupakan fungsi genap, ganjil, atau bukan keduanya? Kemudian sketsakan grafiknya.

Determine whether the rational function f(x) = (2x + 1) / (x - 1) is even, odd, or neither. Learn about vertical and horizontal asymptotes and sketch the graph. This analysis involves key mathematical concepts like asymptotes, even/odd functions, and graphing.

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