The most accurate representations of the domain for the function representing hours slept in one day are B - 0 ≤ x ≤ 24 and E - [0, 24]. These capture the typical and realistic range of sleep duration, excluding invalid extreme values.
Selecting the Correct Representation of the Domain:
A function's domain specifies all possible input values it can accept. In this case, the function takes the number of hours slept in one day as input. Let's analyze each representation:
A - x: This set includes all values greater than 24. While someone might occasionally sleep over 24 hours (due to medical conditions or travel across time zones), it's not a typical daily occurrence. So, this representation is not accurate.
B - 0 ≤ x ≤ 24: This set includes all values between 0 and 24, inclusive. This correctly captures the typical range of sleep duration in a day, making it a valid representation.
C - (24, ∞0): This set excludes 24 and includes all values greater than it. This implies someone can't sleep for 24 hours, which isn't true. Thus, this representation is not accurate.
D - [0, ∞): This set includes all values from 0 onwards. While it covers the typical sleep range, it also includes unrealistic values like sleeping for hundreds of hours. Hence, it's not the most precise representation.
E - [0, 24]: This set precisely captures the typical range of sleep duration in a day, from 0 to 24 hours. It excludes both unrealistic scenarios like sleeping over 24 hours and not sleeping at all. Therefore, this is the most accurate and preferred representation of the domain.
F - X: This is a generic set where x can be any value less than or equal to R. We still need to specify R as the maximum sleep duration in a day for this to be relevant. So, if R = 24, it becomes valid, but the notation is less specific than E.
G - (0, 24): This set excludes 0 hours of sleep, implying it's not possible. However, some people might not sleep at all on rare occasions due to various factors. Therefore, excluding 0 makes this representation less accurate than E.
The most accurate and appropriate representations of the domain are:
B - x
E - [0, 24]
Complete question:
Select all representations for the domain of a function given by the number of hours slept in one day.
A - x
B - x
C - (24, ∞0)
D - [0, ∞)
E - [0, 24]
F - X
G - (0, 24)