Question

select all the intervals where ( h ) is decreasing.

( y = h(x) )

choose all answers that apply:
a ( -4.5 < x < -4 )
b ( -3.5 < x < -2.5 )
c ( 4 < x < 4.5 )
d none of the above

Explanation:

To determine where the function \( h(x) \) (represented by \( y = h(x) \)) is decreasing, we analyze the graph's slope: a function is decreasing when as \( x \) increases, \( y \) decreases.

Step 1: Analyze Option A (\(-4.5 < x < -4\))

On the graph, for \( x \) values between \(-4.5\) and \(-4\), as \( x \) increases (moves from left to right), the \( y \)-values decrease (the graph is falling). So \( h(x) \) is decreasing here.

Step 2: Analyze Option B (\(-3.5 < x < -2.5\))

For \( x \) between \(-3.5\) and \(-2.5\), as \( x \) increases, the \( y \)-values are still decreasing (the graph is at the "bottom" part of the left curve, continuing to fall towards the minimum). Wait, no—wait, the left part of the graph: from \( x < -3 \) (approx), the graph decreases until a minimum (around \( x = -2 \) or so), then increases. Wait, re-examining: the left curve starts high, decreases to a minimum (around \( x = -2 \)), then increases. So between \(-3.5\) and \(-2.5\), \( x \) is increasing, and \( y \) is decreasing (since we're moving towards the minimum). Wait, but let's check the graph again. Wait, the left side: at \( x = -4 \), the graph is high; as \( x \) increases to \(-3\), \(-2\), etc., it decreases to a minimum (around \( x = -2 \)), then increases. So \(-3.5\) to \(-2.5\) is within the decreasing interval of the left curve. Wait, but let's check the options. Wait, no—wait, the right curve: after the peak (around \( x = 3 \)), the graph decreases as \( x \) increases beyond \( 3 \).

Wait, no—let's re-express:

- Left curve: from \( x = -\infty \) (or the left end) to \( x = -2 \) (approx minimum), the function is **decreasing** (as \( x \) increases, \( y \) decreases).
- Then, from \( x = -2 \) to \( x = 3 \) (approx peak), the function is **increasing** (as \( x \) increases, \( y \) increases).
- Then, from \( x = 3 \) to \( x = \infty \) (right end), the function is **decreasing** (as \( x \) increases, \( y \) decreases).

Now, check each option:

- Option A: \(-4.5 < x < -4\): within the left decreasing interval (since \( -4.5 < -4 < -2 \)), so \( h(x) \) is decreasing here.
- Option B: \(-3.5 < x < -2.5\): also within the left decreasing interval (\( -3.5 < -2.5 < -2 \)), so \( h(x) \) is decreasing here. Wait, but is that correct? Wait, the minimum is around \( x = -2 \), so from \( x = -4.5 \) to \( x = -2 \), the function is decreasing. So \(-3.5\) to \(-2.5\) is within that, so \( y \) decreases as \( x \) increases.
- Option C: \( 4 < x < 4.5 \): within the right decreasing interval (since \( 4 > 3 \), and after the peak, the function decreases). So as \( x \) increases from \( 4 \) to \( 4.5 \), \( y \) decreases.

Wait, but let's check the graph again. The user's graph:

- Left side: starts high, goes down to a minimum (around \( x = -2 \)), then up to a peak (around \( x = 3 \)), then down.

So:

- For \( x \) in \(-4.5 < x < -4\): \( x \) increases, \( y \) decreases (A is correct).
- For \( x \) in \(-3.5 < x < -2.5\): \( x \) increases, \( y \) decreases (B is correct? Wait, no—wait, if the minimum is at \( x = -2 \), then from \( x = -4 \) to \( x = -2 \), the function is decreasing. So \(-3.5\) to \(-2.5\) is within that, so \( y \) decreases as \( x \) increases. So B is also correct? Wait, but maybe I made a mistake. Wait, let's check the options again. Wait, the options:

A: \(-4.5 < x < -4\): yes, decreasing.
B: \(-3.5 < x < -2.5\): yes, decreasing (since moving from \( -3.5 \) to \( -2.5 \), \( x \) increases, \( y \) decreases).
C: \( 4 < x < 4.5 \): yes, decreasing (since after the peak, \( x > 3 \), \( y \) decreases as \( x \) increases).

Wait, but maybe the graph's minimum is around \( x = -2 \), so from \( x = -\infty \) to \( x = -2 \), decreasing; \( x = -2 \) to \( x = 3 \), increasing; \( x = 3 \) to \( \infty \), decreasing. So all three intervals (A, B, C) are within decreasing intervals. But let's check the options again. Wait, the user's options:

A: \(-4.5 < x < -4\) – yes, decreasing.
B: \(-3.5 < x < -2.5\) – yes, decreasing (since \( x \) increases, \( y \) decreases).
C: \( 4 < x < 4.5 \) – yes, decreasing (since \( x \) increases, \( y \) decreases).

But maybe I misread the graph. Wait, the left curve: at \( x = -4 \), the graph is high; at \( x = -3 \), lower; at \( x = -2 \), minimum. So from \( x = -4.5 \) to \( x = -4 \), \( x \) increases, \( y \) decreases (A is correct). From \( x = -3.5 \) to \( x = -2.5 \), \( x \) increases, \( y \) decreases (B is correct). From \( x = 4 \) to \( x = 4.5 \), \( x \) increases, \( y \) decreases (C is correct). But maybe the options are designed so that A, B, C are all correct? Wait, but let's confirm.

Wait, the key is: a function is decreasing on an interval if, for any two points \( x_1 < x_2 \) in the interval, \( h(x_1) > h(x_2) \).

- For A: \( x_1 = -4.5 \), \( x_2 = -4 \) (since \( -4.5 < -4 \)). \( h(-4.5) > h(-4) \) (since the graph is falling), so decreasing.
- For B: \( x_1 = -3.5 \), \( x_2 = -2.5 \) (since \( -3.5 < -2.5 \)). \( h(-3.5) > h(-2.5) \) (since moving towards the minimum), so decreasing.
- For C: \( x_1 = 4 \), \( x_2 = 4.5 \) (since \( 4 < 4.5 \)). \( h(4) > h(4.5) \) (since the graph is falling after the peak), so decreasing.

Thus, all three intervals (A, B, C) are where \( h(x) \) is decreasing. Wait, but the options include A, B, C, and D. So the correct answers are A, B, C? Wait, but maybe the graph is different. Wait, the original graph: the left side has a minimum around \( x = -2 \), then increases to a peak around \( x = 3 \), then decreases. So:

- Decreasing intervals: \( (-\infty, -2) \) and \( (3, \infty) \).

So:

- A: \(-4.5 < x < -4\) is within \( (-\infty, -2) \) – decreasing.
- B: \(-3.5 < x < -2.5\) is within \( (-\infty, -2) \) – decreasing.
- C: \( 4 < x < 4.5 \) is within \( (3, \infty) \) – decreasing.

Thus, A, B, and C are all correct.

Answer:

A. \(-4.5 < x < -4\), B. \(-3.5 < x < -2.5\), C. \(4 < x < 4.5\)