Question

a college has a financial aid formula that applies to applicants whose familys income is below a certain threshold. the financial aid can be determined by the function f(x)=20000 + 3000x, where x is the number of siblings (including half - siblings and step - siblings) that the applicant has living in their household. find and interpret the given function values and determine an appropriate domain for the function.
answer attempt 1 out of 2
f(-4)=, meaning a qualifying applicant with siblings in their household would get $ in financial aid. this interpretation in the context of the problem.
f(2)=, meaning a qualifying applicant with siblings in their household would get $ in financial aid. this interpretation in the context of the problem.
f(3.5)=, meaning a qualifying applicant with siblings in their household would get $ in financial aid. this interpretation in the context of the problem.
based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Calculate $f(-4)$

Substitute $x = - 4$ into $f(x)=20000 + 3000x$.
$f(-4)=20000+3000\times(-4)=20000 - 12000=8000$. But having - 4 siblings is not possible in the real - world context, so this interpretation is not valid.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)=20000 + 3000x$.
$f(2)=20000+3000\times2=20000 + 6000=26000$. This means a qualifying applicant with 2 siblings in their household would get $26000$ in financial aid, and this interpretation is valid in the context of the problem.

Step3: Calculate $f(3.5)$

Substitute $x = 3.5$ into $f(x)=20000 + 3000x$.
$f(3.5)=20000+3000\times3.5=20000+10500 = 30500$. But the number of siblings must be a non - negative integer, so having 3.5 siblings is not possible in the real - world context, and this interpretation is not valid.

Step4: Determine the domain

Since the number of siblings $x$ must be a non - negative integer (you can't have a negative or non - whole number of siblings in a household), the domain of the function is the set of non - negative integers $\{0,1,2,\cdots\}$.

Answer:

$f(-4)=8000$, meaning a qualifying applicant with - 4 siblings in their household would get $8000$ in financial aid. This interpretation is not valid in the context of the problem.
$f(2)=26000$, meaning a qualifying applicant with 2 siblings in their household would get $26000$ in financial aid. This interpretation is valid in the context of the problem.
$f(3.5)=30500$, meaning a qualifying applicant with 3.5 siblings in their household would get $30500$ in financial aid. This interpretation is not valid in the context of the problem.
The domain of the function is the set of non - negative integers $\{0,1,2,\cdots\}$.