1. **State the problem:** We have the function $f(x) = 10000 + 1000x$ which represents financial aid amount based on the number of siblings $x$ in the household. 2. **Calculate $f(-4)$:** $$f(-4) = 10000 + 1000 \times (-4) = 10000 - 4000 = 6000$$ Interpretation: A qualifying applicant with $-4$ siblings would get 6000 in financial aid. However, having negative siblings does not make sense in this context, so $x = -4$ is not a valid input. 3. **Calculate $f(3)$:** $$f(3) = 10000 + 1000 \times 3 = 10000 + 3000 = 13000$$ Interpretation: A qualifying applicant with 3 siblings would get 13000 in financial aid. This makes sense because the number of siblings is positive. 4. **Calculate $f(2.5)$:** $$f(2.5) = 10000 + 1000 \times 2.5 = 10000 + 2500 = 12500$$ Interpretation: A qualifying applicant with 2.5 siblings would get 12500 in financial aid. Since the number of siblings must be a whole number, 2.5 siblings is not realistic, so this input is not appropriate. 5. **Determine appropriate domain:** Since the number of siblings must be a whole number and cannot be negative, the domain is: $$x \in \{0,1,2,3,\ldots\}$$ meaning $x$ is a non-negative integer. 6. **Does the answer make sense?** - Negative siblings ($x=-4$) do not make sense. - Fractional siblings ($x=2.5$) do not make sense. - Positive whole number siblings ($x=3$) make sense. Therefore, the function is valid and makes sense only for non-negative integers $x$ representing the number of siblings.