To determine which of the given rational functions corresponds to the graph, we analyze the key features of the graph, such as vertical asymptotes, behavior at those asymptotes, and overall shape.

1. Identifying the vertical asymptote

• The graph has a vertical asymptote at $x = 0$, as the function diverges to $+\infty$ on the right and $-\infty$ on the left of $x = 0$.

This indicates that the denominator must have a factor of $x$, as division by zero at $x = 0$ causes this asymptote.

2. Eliminating incorrect options

Let’s evaluate each given function to see if it has an asymptote at $x = 0$.

Option A: $F(x) = \frac{x-1}{x(x-2)}$

• Denominator: $x(x-2)$.
• Vertical asymptotes occur at $x = 0$ and $x = 2$.
• This matches the behavior observed at $x = 0$.
  Thus, Option A remains a candidate.

Option B: $F(x) = \frac{x+2}{x-2}$

• Denominator: $x-2$.
• Vertical asymptote occurs at $x = 2$, not at $x = 0$.
  Option B is eliminated.

Option C: $F(x) = \frac{x+2}{x+1}$

• Denominator: $x+1$.
• Vertical asymptote occurs at $x = -1$, not at $x = 0$.
  Option C is eliminated.

Option D: $F(x) = \frac{(x+1)}{(x+2)(x-2)}$

• Denominator: $(x+2)(x-2)$.
• Vertical asymptotes occur at $x = -2$ and $x = 2$, not at $x = 0$.
  Option D is eliminated.

3. Verifying Option A

For $F(x) = \frac{x-1}{x(x-2)}$:

• Vertical asymptotes: $x = 0$ and $x = 2$.
• The graph shows a vertical asymptote at $x = 0$, which matches the function.

Furthermore, the behavior of the graph near $x = 0$ (approaching $+\infty$ and $-\infty$) aligns with the properties of Option A.

Final Answer:

The correct function is Option A: $F(x) = \frac{x-1}{x(x-2)}$.

Would you like a detailed explanation of vertical asymptotes or behavior near these points? Here are 5 related questions to explore further:

1. What are vertical and horizontal asymptotes, and how are they determined?
2. How do you analyze the behavior of a rational function near a vertical asymptote?
3. How can you find the domain of a rational function?
4. What is the difference between vertical asymptotes and removable discontinuities?
5. How does the degree of the numerator and denominator affect the graph of a rational function?

Tip: Vertical asymptotes occur where the denominator equals zero, but the numerator is non-zero at that point.