Question

which statement is true about the graphed function?
f(x) < 0 over the interval (-∞, -4)
f(x) < 0 over the interval (-∞, -3)
f(x) > 0 over the interval (-∞, -3)
f(x) > 0 over the interval (-∞, -4)

Explanation:

Step1: Analyze the graph's position

The graph of the function: For \( F(x) > 0 \), the graph is above the \( x \)-axis; for \( F(x) < 0 \), it's below. The graph has a point at \( x = - 4 \) on the \( x \)-axis. For \( x < - 4 \), we check the graph's position. From the graph, to the left of \( x=-4 \) (i.e., \( (-\infty, - 4) \)), the graph is above the \( x \)-axis? Wait, no, wait. Wait, looking at the graph: the curve at \( x < - 4 \): wait, the graph has a vertical line? No, the graph is a curve. Wait, the point at \( x=-4 \) is on the \( x \)-axis. Then, for \( x < - 4 \), let's see the \( y \)-values. Wait, maybe I misread. Wait, the graph: when \( x < - 4 \), what's the \( y \)-value? Wait, the graph is above or below? Wait, the curve: at \( x=-4 \), it's on the \( x \)-axis. Then, for \( x < - 4 \), let's check the graph. Wait, the left side of \( x = - 4 \): the graph is above the \( x \)-axis? Wait, no, maybe I made a mistake. Wait, the options: let's check each interval.

First, interval \( (-\infty, - 4) \): what's \( F(x) \) here? The graph at \( x < - 4 \): looking at the grid, the \( y \)-axis: the graph to the left of \( x=-4 \) (like \( x=-5 \)): is the graph above or below? Wait, the graph has a point at \( x=-4 \) (on x-axis), and to the left of \( x=-4 \), the graph is above the x-axis? Wait, no, maybe the graph is below? Wait, no, the curve: let's see the graph. Wait, the graph is a parabola-like? Wait, the graph has a minimum? Wait, no, the graph: at \( x=-4 \), it's on the x-axis. Then, for \( x < - 4 \), let's take a test point, say \( x=-5 \). What's \( F(-5) \)? From the graph, the left side of \( x=-4 \): the graph is above the x-axis? Wait, no, maybe I got it wrong. Wait, the options:

Option 1: \( F(x) < 0 \) over \( (-\infty, - 4) \): no, because if left of \( x=-4 \) is above x-axis, then \( F(x) > 0 \). Wait, wait, maybe I misread the graph. Wait, the graph: the curve at \( x=-4 \) is on the x-axis. Then, to the left of \( x=-4 \) ( \( x < - 4 \) ), the graph is above the x-axis (so \( F(x) > 0 \)), and to the right of \( x=-4 \) (but before the minimum) is below? Wait, no, the graph: let's look at the y-axis. The y-axis has negative values? Wait, the graph: the bottom part is at \( y=-12 \)? Wait, no, the grid: the y-axis, the numbers: 3, 0, -3, -6, -9, -12? Wait, maybe the y-axis is labeled with negative numbers below the x-axis. So the x-axis is at \( y=0 \). Then, the graph: at \( x=-4 \), it's on \( y=0 \). To the left of \( x=-4 \) ( \( x < - 4 \) ), the graph is above the x-axis ( \( y > 0 \) ), so \( F(x) > 0 \) there. To the right of \( x=-4 \) (but before the minimum), the graph goes down below the x-axis ( \( y < 0 \) ), then up, etc. Wait, but let's check the options:

Option 4: \( F(x) > 0 \) over \( (-\infty, - 4) \): is that true? Let's check the other options.

Option 1: \( F(x) < 0 \) over \( (-\infty, - 4) \): false, because left of \( x=-4 \) is above x-axis.

Option 2: \( F(x) < 0 \) over \( (-\infty, - 3) \): but \( (-\infty, - 3) \) includes \( (-\infty, - 4) \) and \( (-4, - 3) \). In \( (-4, - 3) \), the graph is below x-axis (since at \( x=-4 \) it's on x-axis, then goes down), so \( F(x) < 0 \) in \( (-4, - 3) \), but in \( (-\infty, - 4) \), \( F(x) > 0 \). So \( (-\infty, - 3) \) has both positive (left of -4) and negative (between -4 and -3) parts, so \( F(x) < 0 \) over \( (-\infty, - 3) \) is false.

Option 3: \( F(x) > 0 \) over \( (-\infty, - 3) \): false, because between -4 and -3, \( F(x) < 0 \).

Option 4: \( F(x) > 0 \) over \( (-\infty, - 4) \): true, because left of \( x=-4 \), the graph is above x-axis ( \( y > 0 \) ), so \( F(x) > 0 \) here.

Wait, maybe I made a mistake. Wait, let's re-examine the graph. The graph at \( x=-4 \) is on the x-axis ( \( y=0 \) ). To the left of \( x=-4 \) (e.g., \( x=-5 \) ), the graph is above the x-axis (so \( y > 0 \) ), so \( F(x) > 0 \) for \( x < - 4 \). So the interval \( (-\infty, - 4) \) has \( F(x) > 0 \), so option 4 is correct.

Step2: Verify each option

- Option 1: \( F(x) < 0 \) over \( (-\infty, - 4) \): False, as \( F(x) > 0 \) here.
- Option 2: \( F(x) < 0 \) over \( (-\infty, - 3) \): False, since \( (-\infty, - 4) \) has \( F(x) > 0 \).
- Option 3: \( F(x) > 0 \) over \( (-\infty, - 3) \): False, as \( (-4, - 3) \) has \( F(x) < 0 \).
- Option 4: \( F(x) > 0 \) over \( (-\infty, - 4) \): True, as the graph is above x - axis for \( x < - 4 \).

Answer:

\( F(x) > 0 \) over the interval \( (-\infty, -4) \) (the fourth option: \( F(x) > 0 \) over the interval \( (-\infty, -4) \))