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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!

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### 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.

Find the formula for f, where the graph of the function suggests a logarithmic form.

Learn how to find the formula for a logarithmic function based on its graph, using an integer base and understanding reflections. This problem involves logarithmic transformations and the syntax for inputting logs in online systems. Suitable for Grades 11-12.

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