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A college has a financial aid formula that applies to applicants whose family's income is below a certain threshold. The financial aid can be determined by the function [tex]f(x) = 10000 + 5000x[/tex], where [tex]x[/tex] 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.

1. [tex]f(-2) = \square[/tex], meaning a qualifying applicant with [tex]\square[/tex] siblings in their household would get [tex]\$\square[/tex] in financial aid. This interpretation [tex]\square[/tex] in the context of the problem.
2. [tex]f(2) = \square[/tex], meaning a qualifying applicant with [tex]\square[/tex] siblings in their household would get [tex]\$\square[/tex] in financial aid. This interpretation [tex]\square[/tex] in the context of the problem.
3. [tex]f(2.5) = \square[/tex], meaning a qualifying applicant with [tex]\square[/tex] siblings in their household would get [tex]\$\square[/tex] in financial aid. This interpretation [tex]\square[/tex] in the context of the problem.



Answer :

GinnyAnswer
To address the question, let's evaluate and interpret the function values step-by-step:

1. Evaluate [tex]\( f(-2) \)[/tex]:

[tex]\( f(-2) = 10000 + 5000 \times (-2) = 10000 - 10000 = 0 \)[/tex]

Interpretation: [tex]\( f(-2) = 0 \)[/tex], meaning a qualifying applicant with [tex]\(-2\)[/tex] siblings in their household would get \[tex]$0 in financial aid. This interpretation does not make sense in the context of the problem because the number of siblings cannot be negative. 2. Evaluate \( f(2) \): \( f(2) = 10000 + 5000 \times 2 = 10000 + 10000 = 20000 \) Interpretation: \( f(2) = 20000 \), meaning a qualifying applicant with 2 siblings in their household would get \$[/tex]20000 in financial aid. This interpretation makes sense in the context of the problem.

3. Evaluate [tex]\( f(2.5) \)[/tex]:

[tex]\( f(2.5) = 10000 + 5000 \times 2.5 = 10000 + 12500 = 22500 \)[/tex]

Interpretation: [tex]\( f(2.5) = 22500 \)[/tex], meaning a qualifying applicant with 2.5 siblings in their household would get \[tex]$22500 in financial aid. This interpretation does not make sense because the number of siblings cannot be a fraction. Given these evaluations, let's determine an appropriate domain for the function: - The number of siblings must be a non-negative integer since one cannot have a negative number or a fractional number of siblings. Thus, the appropriate domain for this function is all non-negative integers (0, 1, 2, 3, ...). Summarizing the interpretations and the domain: - \( f(-2) = 0 \), meaning a qualifying applicant with \(-2\) siblings in their household would get \$[/tex]0 in financial aid. This interpretation does not make sense in the context of the problem.
- [tex]\( f(2) = 20000 \)[/tex], meaning a qualifying applicant with 2 siblings in their household would get \[tex]$20000 in financial aid. This interpretation makes sense in the context of the problem. - \( f(2.5) = 22500 \), meaning a qualifying applicant with 2.5 siblings in their household would get \$[/tex]22500 in financial aid. This interpretation does not make sense because the number of siblings cannot be a fraction.

The appropriate domain for this function is all non-negative integers (0, 1, 2, ...).

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