which graph represents the function $f(x)=\frac{2}{x - 1}+4$?

Explanation:

Step1: Identify vertical asymptote

Set the denominator of the fraction equal to zero. For $y = \frac{2}{x - 1}+4$, we have $x-1=0$, so $x = 1$ is the vertical asymptote.

Step2: Identify horizontal asymptote

As $x\to\pm\infty$, the term $\frac{2}{x - 1}\to0$. So $y=4$ is the horizontal asymptote.

Step3: Analyze behavior near asymptotes

When $x$ approaches $1$ from the left ($x\lt1$ and $x\to1$), $x - 1\to0^-$ and $\frac{2}{x - 1}\to-\infty$, so $y=\frac{2}{x - 1}+4\to-\infty$. When $x$ approaches $1$ from the right ($x\gt1$ and $x\to1$), $x - 1\to0^+$ and $\frac{2}{x - 1}\to+\infty$, so $y=\frac{2}{x - 1}+4\to+\infty$.

Answer:

The graph with vertical asymptote $x = 1$ and horizontal asymptote $y = 4$ and the appropriate behavior near $x = 1$ as described above.