To determine the best statement that represents the domain of the function [tex]\( j(t) \)[/tex], we need to understand what the domain represents in this context. The domain of a function is the set of all possible input values (in this case, the variable [tex]\( t \)[/tex]) for which the function is defined.
Juanita deposits $500 into a savings account that earns 1.5% interest annually, compounded once per year. The function [tex]\( j(t) \)[/tex] represents the amount of money in her account [tex]\( t \)[/tex] years after opening the account.
### Step-by-Step:
1. Understanding [tex]\( t \)[/tex]:
- [tex]\( t \)[/tex] represents the number of years after she has made the initial deposit.
- Since time cannot be negative, [tex]\( t \)[/tex] must be non-negative.
- Additionally, since interest is compounded annually, [tex]\( t \)[/tex] should be measured in whole years (0, 1, 2, etc.).
2. Constraints on [tex]\( t \)[/tex]:
- [tex]\( t \geq 0 \)[/tex] because she cannot have opened the account in negative time.
- [tex]\( t \)[/tex] should be a whole number because the compounding happens annually (we can't have fractional years for our input).
Now, let's analyze the given options:
1. [tex]\( t \geq 500 \)[/tex]:
- This is incorrect because it incorrectly implies [tex]\( t \)[/tex] needs to be at least 500 years, which doesn't make sense for the problem.
2. the set of integers:
- While positive integers could be part of the domain, this includes negative integers which do not make sense in this context because [tex]\( t \)[/tex] cannot be negative.
3. [tex]\( t \geq 500 + 500 \cdot 1.5 \%\)[/tex]:
- This suggests a specific starting value for [tex]\( t \)[/tex] that depends on the interest calculation, which is irrelevant to defining the valid input for [tex]\( t \)[/tex]. The function's domain isn't defined by the specific interest-based amount but by possible values of [tex]\( t \)[/tex].
4. the set of whole numbers:
- This is correct. It accurately states that [tex]\( t \)[/tex] must be 0 or any positive whole number (1, 2, 3, etc.).
### Conclusion:
The statement that best represents the domain of the function [tex]\( j(t) \)[/tex] is:
The set of whole numbers.
This means [tex]\( t \)[/tex] can be any non-negative integer (0, 1, 2, 3, ...).
