To solve the problem step by step, let's evaluate the function [tex]\( f(x) = 10(1.3)^{-x} \)[/tex] at the given points and interpret the results in the context of heating costs.
1. Evaluate [tex]\( f(-7) \)[/tex]:
- When [tex]\( x = -7 \)[/tex]:
[tex]\[
f(-7) = 10 \cdot (1.3)^{-(-7)} = 10 \cdot (1.3)^7
\][/tex]
After performing the calculation, we find:
[tex]\[
f(-7) \approx 62.75
\][/tex]
Interpretation: When the low temperature outside is [tex]\(-7^\circ C\)[/tex], it would cost [tex]\(\$62.75\)[/tex] to heat the building.
2. Evaluate [tex]\( f(3) \)[/tex]:
- When [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 10 \cdot (1.3)^{-3}
\][/tex]
After performing the calculation, we find:
[tex]\[
f(3) \approx 4.55
\][/tex]
Interpretation: When the low temperature outside is [tex]\(3^\circ C\)[/tex], it would cost [tex]\(\$4.55\)[/tex] to heat the building.
3. Evaluate [tex]\( f(3.5) \)[/tex]:
- When [tex]\( x = 3.5 \)[/tex]:
[tex]\[
f(3.5) = 10 \cdot (1.3)^{-3.5}
\][/tex]
After performing the calculation, we find:
[tex]\[
f(3.5) \approx 3.99
\][/tex]
Interpretation: When the low temperature outside is [tex]\(3.5^\circ C\)[/tex], it would cost [tex]\(\$3.99\)[/tex] to heat the building.
4. Determine the domain:
The domain refers to all possible values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) \)[/tex] is defined. Since the expression [tex]\( 10(1.3)^{-x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex], the appropriate domain for the function [tex]\( f(x) \)[/tex] is:
- The domain of the function is all real numbers.
Summarizing the interpreted results:
[tex]\[
f(-7) = 62.75, \quad \text{meaning when the low temperature outside is } -7^\circ C, \text{ it would cost } \$62.75 \text{ to heat the building.}
\][/tex]
[tex]\[
f(3) = 4.55, \quad \text{meaning when the low temperature outside is } 3^\circ C, \text{ it would cost } \$4.55 \text{ to heat the building.}
\][/tex]
[tex]\[
f(3.5) = 3.99, \quad \text{meaning when the low temperature outside is } 3.5^\circ C, \text{ it would cost } \$3.99 \text{ to heat the building.}
\][/tex]
The domain of the function is all real numbers.
