The given function is:

$f(x) = \frac{2}{x - 1} + 3$

To analyze and match the correct graph, let's identify the key characteristics of the function:

1. Vertical Asymptote:

The denominator of the fraction is $x - 1$. Setting this equal to zero gives the vertical asymptote: $x - 1 = 0 \implies x = 1$

2. Horizontal Asymptote:

As $x$ approaches infinity or negative infinity, the fraction $\frac{2}{x - 1}$ approaches zero. Thus, the horizontal asymptote is: $f(x) \to 3 \quad \text{(from the "+3" in the equation)}.$

3. Behavior:

• When $x \to 1^+$, the term $\frac{2}{x - 1}$ becomes very large ($+\infty$), so $f(x) \to +\infty$.
• When $x \to 1^-$, the term $\frac{2}{x - 1}$ becomes very negative ($-\infty$), so $f(x) \to -\infty$.

4. Shifts and Orientation:

This is a rational function shifted to the right by 1 unit (due to $x - 1$) and shifted up by 3 units (due to $+3$).

Matching to Graphs:

• The correct graph must have:
  • A vertical asymptote at $x = 1$,
  • A horizontal asymptote at $y = 3$,
  • The curve approaching $-\infty$ on the left side of $x = 1$ and $+\infty$ on the right side.

Answer: The third graph (bottom-left) matches the function $f(x) = \frac{2}{x - 1} + 3$. It shows the correct asymptotes and behavior.

Would you like a step-by-step explanation or visual breakdown of how these properties affect the graph?

Here are 5 related questions for further exploration:

1. What changes in the graph if the fraction changes to $\frac{-2}{x - 1}$?
2. How would the graph differ if the vertical shift (+3) were removed?
3. What happens to the vertical asymptote if the denominator is $x + 2$ instead of $x - 1$?
4. Can you identify the domain and range of this function?
5. How does modifying the numerator (e.g., changing $2$ to $4$) affect the graph?

Tip: Always start analyzing rational functions by identifying the vertical and horizontal asymptotes—they dictate the overall shape of the graph.