Question

passengers
question 4
consider the table below.

x	y
-3	-6
-2	0
-1	4
0.5	6.25
2	4

on which domain does the function’s range increase?
(0.5,2)

Explanation:

Step1: Analyze x and y values

Check the change in \( y \) as \( x \) increases. For \( x = -3, y = -6 \); \( x = -2, y = 0 \) (increase); \( x = -1, y = 4 \) (increase); \( x = 0.5, y = 6.25 \) (increase); \( x = 2, y = 4 \) (decrease). Now check intervals:
- From \( x = -3 \) to \( x = -1 \): \( y \) goes from -6 to 4 (increasing).
- From \( x = -1 \) to \( x = 0.5 \): \( y \) goes from 4 to 6.25 (increasing). Wait, but the option given is \( (0.5, 2) \). Wait, no—wait, when \( x \) increases from -3 to -1, \( y \) increases. Wait, maybe the function is a quadratic? Let's check the symmetry. The \( y \)-values at \( x = -1 \) and \( x = 2 \) are both 4? Wait, no, \( x = -1 \) and \( x = 2 \): wait, \( x = -1 \), \( y = 4 \); \( x = 2 \), \( y = 4 \). So the vertex is around \( x = \frac{-1 + 2}{2}= 0.5 \)? Wait, \( x = 0.5 \), \( y = 6.25 \), which is the maximum? Wait, no—if \( x \) increases from -3 to 0.5, \( y \) increases (from -6 to 6.25), and from 0.5 to 2, \( y \) decreases (from 6.25 to 4). Wait, but the option is \( (0.5, 2) \)—no, that's decreasing. Wait, maybe I misread. Wait, the question is "on which domain does the function’s range increase". Wait, maybe the correct interval is where as \( x \) increases, \( y \) increases. Let's list the \( x \) and \( y \):

\( x: -3, -2, -1, 0.5, 2 \)

\( y: -6, 0, 4, 6.25, 4 \)

So when \( x \) goes from -3 to -1: \( x \) increases, \( y \) increases (from -6 to 4). From -1 to 0.5: \( x \) increases, \( y \) increases (4 to 6.25). From 0.5 to 2: \( x \) increases, \( y \) decreases (6.25 to 4). Wait, but the option given is \( (0.5, 2) \)—that's decreasing. Wait, maybe the question has a typo, or I misread. Wait, no—wait, maybe the interval is \( (-3, -1) \) or \( (-2, 0.5) \), but the option is \( (0.5, 2) \)? No, that's decreasing. Wait, maybe the original table has a different \( x = 1 \) instead of 0.5? Wait, no. Wait, maybe the function is a quadratic with vertex at \( x = 0.5 \), so the function increases on the left of the vertex (domain \( (-\infty, 0.5) \)) and decreases on the right (\( (0.5, \infty) \)). But the option is \( (0.5, 2) \), which is on the right, so \( y \) should decrease. Wait, maybe the question is "decrease", but it says "increase". Wait, maybe I made a mistake. Wait, let's check the \( y \)-values again:

At \( x = -3 \): -6

\( x = -2 \): 0 (increase from -6)

\( x = -1 \): 4 (increase from 0)

\( x = 0.5 \): 6.25 (increase from 4)

\( x = 2 \): 4 (decrease from 6.25)

So the function increases when \( x \) is in \( (-\infty, 0.5) \), because as \( x \) increases towards 0.5, \( y \) increases. So the domain where \( y \) increases is when \( x \) is less than 0.5. But the option given is \( (0.5, 2) \), which is where \( y \) decreases. Wait, maybe the question is reversed? Or maybe the table has a different \( x = 1 \) instead of 2? Wait, no. Alternatively, maybe the user made a typo, but assuming the option is \( (0.5, 2) \) is incorrect, but wait—wait, maybe I misread the \( x = 2 \) value. Wait, the table says \( x = 2 \), \( y = 4 \); \( x = 0.5 \), \( y = 6.25 \). So from 0.5 to 2, \( x \) increases, \( y \) decreases. So the interval where \( y \) increases is, for example, \( (-3, 0.5) \). But the option given is \( (0.5, 2) \)—maybe the question is wrong, but if we have to choose from the option, maybe the intended interval is where \( x \) increases and \( y \) increases, but in the option \( (0.5, 2) \), \( y \) decreases. Wait, maybe the function is not quadratic. Wait, no—wait, maybe the user's table has a typo, but the option is \( (0.5, 2) \), but that's decreasing. Wait, maybe I made a mistake. Alternatively, maybe the question is "on which domain does the function’s range decrease", but it says "increase". Alternatively, maybe the correct interval is \( (-3, -1) \) or \( (-2, 0.5) \), but since the option is \( (0.5, 2) \), maybe the answer is that, but that's decreasing. Wait, no—maybe I misread the \( y \)-value at \( x = 2 \). Wait, the table says \( x = 2 \), \( y = 4 \); \( x = 0.5 \), \( y = 6.25 \). So from 0.5 to 2, \( y \) goes from 6.25 to 4 (decrease). So the interval where \( y \) increases is when \( x \) is less than 0.5. But the option given is \( (0.5, 2) \), which is decreasing. Wait, maybe the question is wrong, but assuming the option is correct (maybe a mistake in the table), but the process: to find where the function is increasing, we check the slope (change in \( y \) over change in \( x \)).

For \( x \) from -3 to -2: \( \frac{0 - (-6)}{-2 - (-3)} = \frac{6}{1} = 6 \) (positive, increasing)

From -2 to -1: \( \frac{4 - 0}{-1 - (-2)} = \frac{4}{1} = 4 \) (positive, increasing)

From -1 to 0.5: \( \frac{6.25 - 4}{0.5 - (-1)} = \frac{2.25}{1.5} = 1.5 \) (positive, increasing)

From 0.5 to 2: \( \frac{4 - 6.25}{2 - 0.5} = \frac{-2.25}{1.5} = -1.5 \) (negative, decreasing)

So the function is increasing on \( (-\infty, 0.5) \) and decreasing on \( (0.5, \infty) \). So the interval where it's increasing would be, for example, \( (-3, 0.5) \), but the option given is \( (0.5, 2) \), which is decreasing. Wait, maybe the question is "decrease", but it says "increase". Alternatively, maybe the table has \( x = 2 \), \( y = 8 \) instead of 4? No. Alternatively, maybe the option is correct, and I made a mistake. Wait, the question is "on which domain does the function’s range increase"—wait, "range" here might be a typo, maybe "function" (i.e., the function is increasing). So the function is increasing when the derivative (or slope) is positive. So from the slopes, the function is increasing when \( x < 0.5 \), so intervals like \( (-3, 0.5) \), but the option is \( (0.5, 2) \), which is decreasing. Wait, maybe the user's option is wrong, but the process is as above.

Wait, maybe the table has \( x = 2 \), \( y = 8 \), but no. Alternatively, maybe the question is "decrease", and the option is correct. But assuming the question is correct, and the option is \( (0.5, 2) \) is wrong, but the process: to determine where the function is increasing, we check the direction of \( y \) as \( x \) increases. From the table, as \( x \) increases from -3 to 0.5, \( y \) increases; from 0.5 to 2, \( y \) decreases. So the domain where the function is increasing is \( (-\infty, 0.5) \), so any subinterval of that, like \( (-3, 0.5) \), but the option given is \( (0.5, 2) \), which is decreasing. Maybe the question has a typo, but the step-by-step is as above.

Answer:

(0.5, 2) is incorrect for increase; the correct interval for increase is \( (-\infty, 0.5) \), but if forced to choose the given option, there's a mistake. However, following the process, the function increases when \( x < 0.5 \).