The given problem asks to find the vertical asymptotes of the function:

$y = (x + 1) \cdot \ln\left(2 - \frac{1}{x}\right)$

To solve for the vertical asymptotes:

1. Look at the logarithmic function: For the natural logarithm function $\ln$, the argument must be strictly positive. So, we need to solve:

$2 - \frac{1}{x} > 0$

$2 > \frac{1}{x} \quad \Rightarrow \quad x > \frac{1}{2}$

Thus, $x = \frac{1}{2}$ is a critical point.

2. Check for undefined points: The logarithmic function becomes undefined if $x = 0$, because $\frac{1}{x}$ would become infinite.

Thus, the vertical asymptotes occur at:

• $x = 0$ (because the logarithmic term is undefined at this point).
• $x = \frac{1}{2}$ (due to the boundary condition).

Conclusion:

The vertical asymptotes are at $x = 0$ and $x = \frac{1}{2}$. Therefore, the correct answer is d. $x = 0$ and $x = \frac{1}{2}$.

Would you like further explanation or details?

Here are 5 related questions to explore:

1. How do you determine horizontal asymptotes for a given function?
2. How does the natural logarithm function behave near its vertical asymptotes?
3. What happens to the function $y = \ln(x)$ as $x$ approaches 0 from the right?
4. Can a function have multiple vertical asymptotes?
5. What role do asymptotes play in curve sketching?

Tip: When working with logarithmic functions, always check the domain carefully to identify where the argument becomes non-positive, as that will give clues to asymptotes or undefined points.