Question

the piecewise function h(x) is shown on the graph. what is the value of x when h(x) = -3? -7 -1 0 2

Explanation:

Step1: Understand the problem

We need to find the value of \( x \) when \( h(x) = -3 \) from the graph of the piecewise function \( h(x) \).

Step2: Analyze the graph

Look at the graph and find the point where the \( y \)-coordinate (which is \( h(x) \)) is equal to \( -3 \). Then, determine the corresponding \( x \)-coordinate.

From the graph, we can see that when \( h(x) = -3 \), we need to check the part of the piecewise function that reaches \( y = -3 \). Looking at the graph, we observe that the line (or segment) that has \( y = -3 \) corresponds to a certain \( x \)-value. Wait, maybe I made a mistake. Wait, let's re-examine. Wait, the graph: let's see the points. Wait, the left part: from \( x = -3 \) (the blue dot at \( (-3, -7) \)) up to \( x = 0 \) (the dot at \( (0, -1) \))? Wait, no, maybe the right part. Wait, the right part starts at \( x = 1 \) (the blue dot at \( (1, -1) \)) and goes down to \( x = 2 \) (open circle at \( (2, -3) \))? Wait, no, the open circle at \( (2, -3) \) means the function approaches \( -3 \) as \( x \) approaches \( 2 \) from the left, but does not include \( x = 2 \). Wait, maybe the left part. Wait, the left line: from \( (-3, -7) \) to \( (0, -1) \)? Wait, no, the slope: let's calculate the slope. From \( (-3, -7) \) to \( (0, -1) \): slope is \( \frac{-1 - (-7)}{0 - (-3)} = \frac{6}{3} = 2 \). So the equation of that line is \( y - (-7) = 2(x - (-3)) \), so \( y + 7 = 2(x + 3) \), so \( y = 2x + 6 - 7 = 2x - 1 \). Let's check at \( x = -3 \): \( y = 2(-3) - 1 = -7 \), correct. At \( x = 0 \): \( y = -1 \), correct. Then the right part: from \( (1, -1) \) to \( (2, -3) \) (open circle). The slope here is \( \frac{-3 - (-1)}{2 - 1} = \frac{-2}{1} = -2 \). So the equation is \( y - (-1) = -2(x - 1) \), so \( y + 1 = -2x + 2 \), so \( y = -2x + 1 \). Let's check at \( x = 1 \): \( y = -2(1) + 1 = -1 \), correct. At \( x = 2 \): \( y = -4 + 1 = -3 \), but it's an open circle, so \( x = 2 \) is not included. Wait, but we need \( h(x) = -3 \). So when does \( h(x) = -3 \)? Let's check the right part: \( y = -2x + 1 = -3 \). Solve for \( x \): \( -2x + 1 = -3 \) → \( -2x = -4 \) → \( x = 2 \). But the open circle at \( (2, -3) \) means the function does not include \( x = 2 \) for \( y = -3 \). Wait, maybe the left part? Wait, the left part: \( y = 2x - 1 \). Set \( y = -3 \): \( 2x - 1 = -3 \) → \( 2x = -2 \) → \( x = -1 \). Wait, but at \( x = -1 \), what's \( y \)? \( y = 2(-1) - 1 = -3 \). Wait, but is \( x = -1 \) in the domain of the left part? The left part: from \( x = -3 \) (closed dot) to \( x = 0 \) (closed dot)? Wait, the graph shows a closed dot at \( (0, -1) \)? Wait, the original graph: the blue dot at \( (0, -1) \)? Wait, the user's graph: let's look again. The graph has a blue dot at \( (0, -1) \)? Wait, the left line: from \( (-3, -7) \) (blue dot) to \( (0, -1) \) (blue dot)? Then the right line: from \( (1, -1) \) (blue dot) to \( (2, -3) \) (open circle). Wait, maybe I misread the graph. Wait, the problem is to find \( x \) when \( h(x) = -3 \). Let's check the options: -7, -1, 0, 2. Let's check each option:

- For \( x = -7 \): Is there a point at \( x = -7 \)? The graph starts at \( x = -7 \)? No, the leftmost point is \( x = -3 \) (the blue dot at \( (-3, -7) \)). So \( x = -7 \) is not on the graph.

- For \( x = -1 \): Let's see, if \( x = -1 \), what's \( h(x) \)? Using the left line equation \( y = 2x - 1 \), at \( x = -1 \), \( y = 2(-1) - 1 = -3 \). So \( h(-1) = -3 \). Wait, but is \( x = -1 \) in the domain of the left part? The left part goes from \( x = -3 \) to \( x = 0 \) (since the dot at \( (0, -1) \) is closed). So \( x = -1 \) is between -3 and 0, so it's included. So \( h(-1) = -3 \). Wait, but the options include -1. Wait, but let's check the graph again. Wait, the graph: the left line (from \( (-3, -7) \) to \( (0, -1) \)) passes through \( (-1, -3) \), because when \( x = -1 \), \( y = 2(-1) - 1 = -3 \). So that's correct. So \( x = -1 \) when \( h(x) = -3 \). Wait, but the options have -1 as an option (second option: -1). So that's the answer.

Wait, but let's confirm with the graph. If we look at the graph, the point \( (-1, -3) \) should be on the left line. So when \( h(x) = -3 \), \( x = -1 \).

Step3: Confirm the answer

Looking at the options, the second option is -1. So the value of \( x \) when \( h(x) = -3 \) is -1.

Answer:

-1 (corresponding to the option: -1)