In this scenario, the function describing the bacterial count is $f(x) = 2.5x + 4$, where $x$ is the number of hours passed.

Finding the Reasonable Range:

1. Understanding the Mathematical Range: The function is linear, and as $x$ increases, $f(x)$ increases without bound. Therefore, the mathematical range of $f(x)$ is all values greater than or equal to 4 (since when $x = 0$, $f(x) = 4$).

2. Determining the Reasonable Range: In a real experiment, bacterial growth may not increase indefinitely due to factors like limited resources. The reasonable range will start from 4 (the initial amount at $x = 0$) and has an unknown upper limit depending on real-world constraints.

Thus, the reasonable range of the function is: $y \geq 4$

Let me know if you'd like a more detailed breakdown or have additional questions!

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Related Questions to Explore

1. How would the range change if the function were exponential instead of linear?
2. What does the slope (2.5) of the function indicate in the context of bacterial growth?
3. How could environmental factors influence the reasonable range of this function?
4. What would be the domain of this function in a real-world experiment?
5. How could we estimate the upper limit of the reasonable range in a laboratory setting?

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Tip: For realistic modeling, always consider practical constraints that can limit the mathematical predictions of a function.