To determine the range of the function that represents the amount of water remaining in Raj's bathtub over time, we first need to look at the given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 40 \\
\hline
0.5 & 39.25 \\
\hline
1 & 38.5 \\
\hline
1.5 & 37.75 \\
\hline
\end{array}
\][/tex]
Here, [tex]\( x \)[/tex] is the time in minutes, and [tex]\( y \)[/tex] is the amount of water remaining in gallons.
To find the range of the function, we need to identify the minimum and maximum values of [tex]\( y \)[/tex] in the given table. From the table:
- At [tex]\( x = 0 \)[/tex], [tex]\( y = 40 \)[/tex]
- At [tex]\( x = 0.5 \)[/tex], [tex]\( y = 39.25 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( y = 38.5 \)[/tex]
- At [tex]\( x = 1.5 \)[/tex], [tex]\( y = 37.75 \)[/tex]
By inspecting these values, we can see:
- The minimum value of [tex]\( y \)[/tex] is 37.75.
- The maximum value of [tex]\( y \)[/tex] is 40.
Therefore, the amount of water [tex]\( y \)[/tex] is between 37.75 gallons and 40 gallons. Thus, the range of the function is:
[tex]\[
\text{all real numbers such that } 37.75 \leq y \leq 40.
\][/tex]
In conclusion, the correct answer is:
[tex]\[
\boxed{\text{all real numbers such that } 37.75 \leq y \leq 40}
\][/tex]
