To determine which graph matches the function $f(x) = \frac{2x + 2}{x}$, let's first simplify the function and analyze its key characteristics.

Simplification:

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

This is a rational function, and its behavior can be analyzed as follows:

1. Vertical Asymptote: The function has a vertical asymptote where the denominator is zero, which is at $x = 0$.

2. Horizontal Asymptote: As $x \to \infty$ or $x \to -\infty$, the term $\frac{2}{x}$ approaches 0. Thus, the horizontal asymptote is $y = 2$.

3. Intercepts:

   • x-intercept: Set $f(x) = 0$: $2 + \frac{2}{x} = 0 \quad \Rightarrow \quad \frac{2}{x} = -2 \quad \Rightarrow \quad x = -1$
   • y-intercept: Set $x = 1$: $f(1) = 2 + \frac{2}{1} = 4$

Graph Characteristics:

• The graph has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 2$.
• There are two branches:
  • As $x \to 0^+$, $f(x) \to \infty$, and as $x \to 0^-$, $f(x) \to -\infty$.
  • As $x \to \infty$, $f(x) \to 2$ from above, and as $x \to -\infty$, $f(x) \to 2$ from below.

Correct Graph:

• The correct graph matches the description of Option B, which has a vertical asymptote at $x = 0$, a horizontal asymptote at $y = 2$, and two branches (one below $y = 2$ on the left and one above $y = 2$ on the right).

Let me know if you'd like more details on the function analysis or graphing!

Related Questions:

1. How do you determine vertical asymptotes of a rational function?
2. What is the significance of horizontal asymptotes in rational functions?
3. How can we find the x-intercepts and y-intercepts of rational functions?
4. What is the difference between vertical and horizontal asymptotes in terms of limits?
5. How does the function's behavior near asymptotes influence its graph?

Tip: Always check for vertical asymptotes by finding where the denominator of a rational function equals zero.