which graph represents the function $f(x)=\frac{2x - 1}{x - 1}$?

Answer:

The graph with a vertical asymptote at (x = 1), horizontal asymptote at (y = 2), (x)-intercept at (\left(\frac{1}{2}, 0\right)), and (y)-intercept at ((0, 1)), where the function approaches (+\infty) as (x \to 1^+) and (-\infty) as (x \to 1^-), while approaching (y = 2) as (x \to \pm\infty).

Explanation:

Step 1: Identify vertical asymptote

Denominator (x - 1 = 0 \implies x = 1).

Step 2: Find horizontal asymptote

Degree of numerator = degree of denominator, so (y = \frac{2}{1} = 2).

Step 3: Calculate (x)-intercept

Set numerator to 0: (2x - 1 = 0 \implies x = \frac{1}{2}).

Step 4: Calculate (y)-intercept

Evaluate at (x = 0): (f(0) = \frac{-1}{-1} = 1).

Step 5: Analyze behavior around (x = 1)

As (x \to 1^+), (2x - 1 \to 1), (x - 1 \to 0^+ \implies f(x) \to +\infty).
As (x \to 1^-), (2x - 1 \to 1), (x - 1 \to 0^- \implies f(x) \to -\infty).