To address the problem, let's break it down step by step.

Problem Summary:

The given function to determine the daily cost to heat a building based on the outside temperature is: $f(x) = 9(1.05)^x$ where $x$ is the low temperature outside in degrees Celsius, and $f(x)$ is the cost in dollars.

You need to:

1. Find and interpret the given function values.
2. Determine an appropriate domain for the function.

Part 1: Calculate and Interpret Function Values

1. $f(-8)$

$f(-8) = 9(1.05)^{-8}$

2. $f(10)$

$f(10) = 9(1.05)^{10}$

3. $f(5)$

$f(5) = 9(1.05)^5$

I'll calculate these values for you.

Part 2: Appropriate Domain

• The domain of the function depends on realistic temperatures during winter, which are usually between certain negative and positive values.

Let's first calculate the values and then interpret them. I'll do the calculations now.### Calculations:

1. $f(-8) = 6.09$

   • Interpretation: When the outside temperature is $-8^\circ \text{C}$, the cost to heat the building is $6.09.
   • This interpretation makes sense, as colder temperatures typically require more heating, but the base function indicates that for negative temperatures, the cost decreases slightly.

2. $f(10) = 14.66$

   • Interpretation: When the outside temperature is $10^\circ \text{C}$, the cost to heat the building is $14.66.
   • This does not make sense in the context of the problem because it implies that warmer temperatures require more heating, which is illogical.

3. $f(5) = 11.49$

   • Interpretation: When the outside temperature is $5^\circ \text{C}$, the cost to heat the building is $11.49.
   • This makes sense, as moderately cold temperatures would require some heating, but less than more extreme cold.

Conclusion on Domain:

The function seems more reasonable for temperatures around or below freezing, as higher temperatures (like 10°C) should not increase heating costs. Therefore, the appropriate domain for this function is likely to be for $x \leq 0$ or a slightly positive range, such as $x \leq 5$.

Would you like more details on any part, or have any other questions?

Follow-up Questions:

1. Why does the function decrease in cost as the temperature becomes negative?
2. How would the interpretation change if the base cost was different?
3. What would happen to the domain if we used a different base for the exponent?
4. How would the function behave if we included insulation factors?
5. Could this model be used for other types of buildings or regions?

Tip:

Always check the context of mathematical models to ensure the interpretations make practical sense, especially when working with real-world data.