a owes $75. She plans on paying back her debt at $3 per day. The amount she owes after a given number of days can be represented by y=3x−75.
Which of the following correctly analyzes the domain of this situation?
1) The domain represents time, in days, and since time has to be greater than or equal to 0, x≥0.
2)Since there are no restrictions placed on the values of x, the domain is all real numbers.
3)The domain represents the amount owed and since this value has to be less than or equal to 75, x≤75.
4)Since 3x−75 has to be greater than 0, the value of x has to be greater than 25, or x>25.
I chose the answer 1. Is that correct?
1 is only "partly" correct. For large values of days #1 would not be correct because you would have already paid off the entire debt.
What do you think of 2, 3 and 4?
I know it's not 3 because 75 is already in the equation(the amount owed)
I'm not sure about 4 because I have no idea where they go 25
But 2 looks but doesn't look correct because there are a set restrictions on x because x can not be a negative, but it can be all real numbers.
Yes, your choice of answer 1 is correct. Let me explain why.
In this situation, the variable x represents the number of days, and we want to analyze the domain, which represents the possible values for x.
Option 1 states that "the domain represents time, in days, and since time has to be greater than or equal to 0, x ≥ 0." This means that x can take on values starting from 0 and can go up to any positive value, including fractions or decimals of days.
Option 2 suggests that there are no restrictions placed on the values of x, and therefore, the domain is all real numbers. However, in this scenario, it is necessary to consider that x represents the number of days, so it cannot have negative values or any values that are not valid in the context of time.
Option 3 states that "the domain represents the amount owed and since this value has to be less than or equal to 75, x ≤ 75." This option mixes up the concept of the domain and the range. In this situation, x represents the number of days, not the amount owed.
Option 4 states that "since 3x−75 has to be greater than 0, the value of x has to be greater than 25, or x > 25." This option mistakenly assumes that the amount owed has to be greater than $75, while the equation y = 3x - 75 actually indicates the remaining amount owed after a given number of days. The domain, however, pertains to the possible values for x.
Therefore, the correct choice is option 1, which correctly analyzes the domain by stating that x (the number of days) has to be greater than or equal to 0 (since time cannot be negative).