Let's break down the given function:

The function is $f(x) = \log_2(-x) - 2$.

Key features:

1. Logarithmic Nature: The logarithmic function $\log_2(-x)$ is defined only when $-x > 0$, i.e., when $x < 0$. This restricts the domain to negative values of $x$.
2. Vertical Shift: The "-2" in $\log_2(-x) - 2$ means that the entire graph is shifted down by 2 units.
3. Asymptotes: The function $\log_2(-x)$ has a vertical asymptote at $x = 0$ (since $\log_2(-x)$ becomes undefined at $x = 0$).

Behavior:

• The graph is only defined for $x < 0$.
• There is a vertical asymptote at $x = 0$, meaning the graph will approach negative infinity as $x$ approaches 0 from the left.
• The entire graph is shifted downward by 2 units because of the "-2".

Graph Selection:

• The correct graph should:
  • Only exist on the negative side of the x-axis.
  • Have a vertical asymptote at $x = 0$.
  • Be shifted downward by 2 units.

Looking at the graphs provided, the third graph (from top to bottom) fits this description: it is on the negative side, has an asymptote at $x = 0$, and the shift downwards by 2 is visible.

Let me know if you would like any further explanation!

Here are 5 relative questions:

1. How would the graph change if we had $f(x) = \log_2(x) - 2$?
2. What happens if we change the base of the logarithm, such as $f(x) = \log_3(-x) - 2$?
3. How would the graph behave if the vertical shift was positive, like $f(x) = \log_2(-x) + 2$?
4. What is the domain and range of $f(x) = \log_2(-x) - 2$?
5. How can we determine the asymptotes of a logarithmic function?

Tip: When analyzing transformations of functions, always consider shifts (vertical and horizontal) and stretches to understand the overall change in the graph.