To solve the problem, let’s go step by step based on the given image:

1. Transformations

The function given is $h(x) = -\ln(x)$. The base function is $f(x) = \ln(x)$.

• The negative sign in front of the logarithm ($-$) reflects the graph of $\ln(x)$ about the x-axis. Therefore, the correct choice is: B. The graph of $f(x) = \ln(x)$ should be reflected about the x-axis.

2. Equation of the Vertical Asymptote

The logarithmic function $\ln(x)$ is undefined for $x \leq 0$, meaning the vertical asymptote for $h(x)$ is at $x = 0$.

Answer: The equation of the vertical asymptote is: $x = 0$

3. Domain of $h(x) = -\ln(x)$

The domain of a logarithmic function is determined by the argument inside the logarithm. Since $h(x)$ involves $\ln(x)$, $x$ must be greater than $0$. Therefore, the domain of $h(x)$ is: $(0, \infty)$

Summary of Answers:

1. Transformation: B
2. Vertical Asymptote: $x = 0$
3. Domain: $(0, \infty)$

Would you like further clarification on graphing or the reasoning behind these answers?

Here are 5 related questions for deeper understanding:

1. How does reflecting a graph about the x-axis affect its points?
2. Why does the vertical asymptote of $\ln(x)$ remain the same after transformations?
3. What is the range of $h(x) = -\ln(x)$?
4. Can we shift $\ln(x)$ horizontally or vertically and how would it affect the asymptote?
5. What happens to the graph of $\ln(x)$ when reflected about the y-axis instead?

Tip: Always identify the transformations step by step (reflection, shifting, stretching) for any function, starting from the base function!