The problem involves analyzing the function $f(x) = \frac{2x+1}{x-1}$.

1. Type of Function: First, let's determine whether $f(x)$ is even, odd, or neither.

   • A function is even if $f(-x) = f(x)$.
   • A function is odd if $f(-x) = -f(x)$.
   • If neither condition is met, the function is neither even nor odd.

2. Asymptotes: Next, we'll look for any vertical or horizontal asymptotes.

   • A vertical asymptote occurs where the denominator is zero, unless the same factor also zeros out the numerator.
   • A horizontal asymptote is determined by the degrees of the numerator and denominator or the behavior of $f(x)$ as $x$ approaches infinity or negative infinity.

3. Graphing: Finally, we'll sketch the graph based on this analysis.

Let's start by identifying if $f(x)$ is even, odd, or neither:

For even: $f(-x) = \frac{2(-x) + 1}{-x - 1} = \frac{-2x + 1}{-x - 1}$ $f(-x) \neq f(x)$

For odd: $f(-x) = \frac{-2x + 1}{-x - 1} = \frac{2x - 1}{x + 1}$ $f(-x) \neq -f(x)$

Thus, $f(x)$ is neither even nor odd.

For the asymptotes:

• Vertical asymptote: Set $x - 1 = 0$, so $x = 1$.
• Horizontal asymptote: Since the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients: $y = \frac{2}{1} = 2$.

Let's graph this function to visualize it better.