Question
if the function f(y) = n models the number, n, of inches a tree grows in a year, y, which describes an appropriate domain for the function? Responses A all integers where y > 0all integers where y > 0 b. all rational numbers where y > 0all rational numbers where y > 0 c. all rational numbers where y ≥ 0
Answers
GPT-4o mini
To determine an appropriate domain for the function \( f(y) = n \), which models the number of inches a tree grows in a year \( y \), we need to consider the nature of the variable \( y \).
1. **Nature of \( y \)**: Since \( y \) represents the number of years, it makes sense for it to be a non-negative integer because you cannot have a negative year or a fraction of a year in this context.
2. **Analysis of Responses**:
- **Response A**: "All integers where \( y > 0 \)" is incorrect because it does not include \( y = 0 \), which could represent the starting point (the initial planting year).
- **Response B**: "All rational numbers where \( y > 0 \)" is incorrect because rational numbers include fractions, which do not apply in the context of counting full years.
- **Response C**: "All rational numbers where \( y \geq 0 \)" is also incorrect since, similar to Response B, it allows for fractions, which are not appropriate in this context.
Given this analysis, the most appropriate domain is actually non-negative integers (since \( y \) should be whole years starting from 0). However, since none of the options precisely matches this standard, if we have to choose among the given choices, the correct phrasing should ideally reflect "non-negative integers" but cannot be quite decided from the options.
If you're looking for the closest option, **Response A** is preferable, as it captures a valid aspect but is still incomplete because it should include \( y = 0 \).
In conclusion, none of the options perfectly fit the context, but we need to consider that **and explicitly state that the best domain should actually be "non-negative integers" \( (y \geq 0) \)**.
1. **Nature of \( y \)**: Since \( y \) represents the number of years, it makes sense for it to be a non-negative integer because you cannot have a negative year or a fraction of a year in this context.
2. **Analysis of Responses**:
- **Response A**: "All integers where \( y > 0 \)" is incorrect because it does not include \( y = 0 \), which could represent the starting point (the initial planting year).
- **Response B**: "All rational numbers where \( y > 0 \)" is incorrect because rational numbers include fractions, which do not apply in the context of counting full years.
- **Response C**: "All rational numbers where \( y \geq 0 \)" is also incorrect since, similar to Response B, it allows for fractions, which are not appropriate in this context.
Given this analysis, the most appropriate domain is actually non-negative integers (since \( y \) should be whole years starting from 0). However, since none of the options precisely matches this standard, if we have to choose among the given choices, the correct phrasing should ideally reflect "non-negative integers" but cannot be quite decided from the options.
If you're looking for the closest option, **Response A** is preferable, as it captures a valid aspect but is still incomplete because it should include \( y = 0 \).
In conclusion, none of the options perfectly fit the context, but we need to consider that **and explicitly state that the best domain should actually be "non-negative integers" \( (y \geq 0) \)**.