It looks like the function you're trying to determine, based on the graph provided, is related to logarithmic functions. You attempted to input $\log_2(-x)$, but you received an error stating "Unexpected character '2'." This suggests that the input syntax for logarithmic functions in the system might not allow direct input of subscript bases, like $\log_2$.

From the graph, it seems the function has characteristics of a logarithmic curve, specifically shifted or reflected along the y-axis, which suggests that it could be of the form:

$f(x) = \log_b(-x)$

Where $b$ is an integer base, which could be 2, given the hint about using an integer base.

To fix the "Unexpected character '2'" error, you might need to input the logarithm in a specific format that the system expects. Try using log(-x) for the natural logarithm, or log(x) with a different syntax for a base (e.g., log(x)/log(2) if you need base 2).

Let me know if you'd like help figuring out the exact input format, or if you'd like more details on logarithmic functions!

Follow-up questions:

1. How do different bases affect the shape of a logarithmic function?
2. Why is the function reflected about the y-axis in this case?
3. What are the common domains of logarithmic functions and how do they change with negative inputs like $-x$?
4. How can logarithmic functions be used to solve real-world problems involving exponential growth?
5. What is the impact of different transformations (translations, reflections) on the graph of a logarithmic function?

Tip: Always check the input syntax for specific mathematical operations in online platforms, as they often have unique requirements for symbols like logarithmic bases.