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x = 0 to x = 2; increasing; x = 2 to x = 4; x = 3 to x = 5; x = 4 to x …

Question

x = 0 to x = 2; increasing; x = 2 to x = 4; x = 3 to x = 5; x = 4 to x = 7; decreasing; x = 0 to x = 4; x = 3 to x = 7; neither increasing nor decreasing

Explanation:

Response

To solve this problem, we need to categorize each interval based on whether the function is increasing, decreasing, or neither. Let's analyze each interval:

1. Increasing Intervals:
  • An interval is increasing if, as \( x \) increases, \( y \) (the function's value) also increases.
  • \( x = 0 \) to \( x = 2 \): If the function rises from \( x = 0 \) to \( x = 2 \), this is increasing.
  • \( x = 2 \) to \( x = 4 \): If the function rises from \( x = 2 \) to \( x = 4 \), this is increasing.
  • \( x = 0 \) to \( x = 4 \): This combines \( x = 0 \) to \( x = 2 \) and \( x = 2 \) to \( x = 4 \), so if both parts are increasing, this is increasing.
  • \( x = 3 \) to \( x = 5 \): If the function rises from \( x = 3 \) to \( x = 5 \), this is increasing.
2. Decreasing Intervals:
  • An interval is decreasing if, as \( x \) increases, \( y \) decreases.
  • \( x = 4 \) to \( x = 7 \): If the function falls from \( x = 4 \) to \( x = 7 \), this is decreasing.
  • \( x = 3 \) to \( x = 7 \): This combines \( x = 3 \) to \( x = 5 \) (if increasing) and \( x = 5 \) to \( x = 7 \) (if decreasing). Wait, no—if from \( x = 3 \) to \( x = 7 \) the function overall decreases, but actually, if \( x = 3 \) to \( x = 5 \) is increasing and \( x = 5 \) to \( x = 7 \) is decreasing, maybe not. Wait, maybe the intended decreasing intervals are \( x = 4 \) to \( x = 7 \) and maybe \( x = 3 \) to \( x = 7 \) if the function decreases from \( x = 3 \) to \( x = 7 \). But let's check the "Neither" category.
3. Neither Increasing nor Decreasing:
  • An interval where the function is constant (horizontal line) or has both increasing and decreasing parts (so it's not strictly increasing or decreasing). But usually, "neither" means the function is constant (no change in \( y \) as \( x \) changes) or has fluctuations. However, in typical problems, if there's no interval given as constant, maybe the "neither" is for intervals where the function isn't strictly increasing or decreasing. But in the given options, the "Neither" box is to be matched with intervals that don't fit increasing or decreasing. Wait, maybe the intended matches are:
Matching:
  • Increasing: \( x = 0 \) to \( x = 2 \), \( x = 2 \) to \( x = 4 \), \( x = 0 \) to \( x = 4 \), \( x = 3 \) to \( x = 5 \)
  • Decreasing: \( x = 4 \) to \( x = 7 \), \( x = 3 \) to \( x = 7 \) (if the function decreases from \( x = 3 \) to \( x = 7 \))
  • Neither: The remaining intervals? Wait, maybe the problem is a drag-and-drop where we match each interval to the category. Let's assume the standard:
  • Increasing: Intervals where \( y \) increases with \( x \).
  • Decreasing: Intervals where \( y \) decreases with \( x \).
  • Neither: Intervals where \( y \) is constant (no change) or has both up and down (but in basic problems, constant is rare, so maybe the "Neither" is for intervals not strictly increasing or decreasing, but here maybe the "Neither" box is to be matched with intervals that don't fit. Wait, maybe the correct matches are:
  • Increasing: \( x = 0 \) to \( x = 2 \), \( x = 2 \) to \( x = 4 \), \( x = 0 \) to \( x = 4 \), \( x = 3 \) to \( x = 5 \)
  • Decreasing: \( x = 4 \) to \( x = 7 \), \( x = 3 \) to \( x = 7 \)
  • Neither: The empty boxes (but since the problem is to match, maybe the "Neither" is for intervals that are constant, but if no constant intervals, maybe the problem has a typo. Alternatively, maybe the intended answers are:
Final Matches (Assuming Typical Problem):
  • Increasing: \( x = 0 \) to \( x = 2 \), \( x = 2 \) to \( x =…

Answer:

  • Increasing: \( x = 0 \) to \( x = 2 \), \( x = 2 \) to \( x = 4 \), \( x = 0 \) to \( x = 4 \), \( x = 3 \) to \( x = 5 \)
  • Decreasing: \( x = 4 \) to \( x = 7 \), \( x = 3 \) to \( x = 7 \)
  • Neither: (No intervals, but the box is present—maybe a mistake.)

But since the problem is a drag-and-drop, the correct matches are:

  • Increasing: \( x = 0 \) to \( x = 2 \), \( x = 2 \) to \( x = 4 \), \( x = 0 \) to \( x = 4 \), \( x = 3 \) to \( x = 5 \)
  • Decreasing: \( x = 4 \) to \( x = 7 \), \( x = 3 \) to \( x = 7 \)
  • Neither: (No intervals, but the box is for intervals that aren't increasing or decreasing—maybe none here.)

Alternatively, maybe the "Neither" is for intervals where the function is constant, but if no such interval, the answer is as above.