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)=10000 + 5000x, 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(-2)=, meaning a qualifying applicant with siblings in their household would get $ in financial aid. this interpretation in the context of the problem.
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(2.5)=, 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 $f(-2)$

Substitute $x = - 2$ into $f(x)=10000 + 5000x$.
$f(-2)=10000+5000\times(-2)=10000 - 10000=0$
This is not valid in context as number of siblings can't be negative.

Step2: Calculate $f(4)$

Substitute $x = 4$ into $f(x)=10000 + 5000x$.
$f(4)=10000+5000\times4=10000 + 20000=30000$
It means a qualifying applicant with 4 siblings in their household would get $30000$ in financial aid. This is valid in context.

Step3: Calculate $f(2.5)$

Substitute $x = 2.5$ into $f(x)=10000 + 5000x$.
$f(2.5)=10000+5000\times2.5=10000+12500 = 22500$
But number of siblings must be a whole - number, so this is not valid in context.

Step4: Determine the domain

The number of siblings $x$ must be a non - negative integer. So the domain is the set of non - negative integers $\{0,1,2,\cdots\}$.

Answer:

$f(-2)=0$, meaning a qualifying applicant with $- 2$ siblings in their household would get $\$0$ in financial aid. This interpretation is not valid in the context of the problem.
$f(4)=30000$, meaning a qualifying applicant with $4$ siblings in their household would get $\$30000$ in financial aid. This interpretation is valid in the context of the problem.
$f(2.5)=22500$, meaning a qualifying applicant with $2.5$ siblings in their household would get $\$22500$ in financial aid. This interpretation is not valid in the context of the problem.
Based on the observations above, it is clear that an appropriate domain for the function is the set of non - negative integers $\{0,1,2,\cdots\}$