To determine the range of the function that represents the amount of water remaining in the bathtub, [tex]\( y \)[/tex], as a function of the time in minutes, [tex]\( x \)[/tex], during which it has been draining at a rate of 1.5 gallons per minute, we need to assess the possible values that [tex]\( y \)[/tex] can take.
1. Understanding the Problem:
- The bathtub is clogged and is draining at a constant rate of 1.5 gallons per minute.
- We are given constraints: [tex]\( y \leq 40 \)[/tex] and [tex]\( y \geq 0 \)[/tex].
2. Identifying the Initial and Final Conditions:
- The maximum amount of water that the bathtub can hold, [tex]\( y \)[/tex], is 40 gallons. This is the initial condition, so [tex]\( y \leq 40 \)[/tex].
- As the bathtub drains, [tex]\( y \)[/tex] decreases until it potentially reaches 0 gallons, which is the minimum condition, so [tex]\( y \geq 0 \)[/tex].
3. Forming the Range of [tex]\( y \)[/tex]:
- Considering the constraints, [tex]\( y \)[/tex] starts from a maximum of 40 gallons when the draining starts.
- It can decrease down to a minimum of 0 gallons as the draining continues.
4. Conclusion:
- Therefore, [tex]\( y \)[/tex] can take any value between 0 and 40 gallons inclusive.
- The range of the function [tex]\( y \)[/tex] is all real numbers such that [tex]\( 0 \leq y \leq 40 \)[/tex].
Thus, the correct range of the function is:
[tex]\[
\text{all real numbers such that } 0 \leq y \leq 40
\][/tex]
