Question

the graph of a function h is shown below. find h(-2) and find one value of x for which h(x)=-2. (a) h(-2)= (b) one value of x for which h(x)=-2:

Explanation:

Step1: Find \( h(-2) \)

To find \( h(-2) \), we look at the graph of the function \( h \) and find the \( y \)-value (output) when \( x = -2 \) (input). From the graph, when \( x = -2 \), we check the corresponding point on the graph. Looking at the \( x \)-axis at \( x = -2 \), the point on the graph has a \( y \)-value. From the graph, it seems that at \( x = -2 \), the function's value (the \( y \)-coordinate) is 3? Wait, no, let's re-examine. Wait, the graph: let's see the \( x \)-values. Wait, the \( x \)-axis is marked with -3, -2, -1, 0, 1, etc. Wait, when \( x = -2 \), what's the \( y \)-value? Wait, maybe I misread. Wait, the graph: let's check the coordinates. Wait, maybe the graph has a point at \( x = -2 \) with \( y = 3 \)? Wait, no, maybe I made a mistake. Wait, let's look again. Wait, the first part: \( h(-2) \). Let's see the graph. When \( x = -2 \), the function's value (the \( y \)-coordinate) is 3? Wait, no, maybe the graph is such that at \( x = -2 \), the \( y \)-value is 3? Wait, maybe I need to check the graph again. Wait, perhaps the graph at \( x = -2 \) is at \( y = 3 \)? Wait, no, maybe I made a mistake. Wait, alternatively, maybe when \( x = -2 \), the \( y \)-value is 3? Wait, no, let's think again. Wait, the problem is about the graph of function \( h \). To find \( h(-2) \), we find the point on the graph where \( x = -2 \), then the \( y \)-coordinate of that point is \( h(-2) \). From the graph, looking at \( x = -2 \), the corresponding \( y \)-value (the height of the graph at \( x = -2 \)) is 3? Wait, no, maybe it's 3? Wait, maybe I'm wrong. Wait, alternatively, maybe the graph at \( x = -2 \) is 3. Wait, but let's check the second part: finding \( x \) where \( h(x) = -2 \). So \( h(x) = -2 \) means we look for the \( x \)-value where the \( y \)-value is -2. From the graph, the lowest point (the minimum) is at \( y = -2 \), and the \( x \)-value there is -1? Wait, no, the graph has a minimum at \( x = -1 \) with \( y = -2 \)? Wait, the graph shows a curve that dips down to \( y = -2 \) at \( x = -1 \)? Wait, the \( x \)-axis: -1 is there. Wait, the minimum point is at \( x = -1 \), \( y = -2 \). So for part (b), one value of \( x \) where \( h(x) = -2 \) is \( x = -1 \). For part (a), \( h(-2) \): when \( x = -2 \), what's the \( y \)-value? Let's see the graph. At \( x = -2 \), the graph is at \( y = 3 \)? Wait, maybe the graph at \( x = -2 \) is 3. Wait, perhaps the first part: \( h(-2) = 3 \), and the second part: \( x = -1 \) (since at \( x = -1 \), \( h(x) = -2 \)). Wait, let's confirm. So for \( h(-2) \), we go to \( x = -2 \) on the \( x \)-axis, then up (or down) to the graph, then left (or right) to the \( y \)-axis. So at \( x = -2 \), the graph's \( y \)-coordinate is 3. So \( h(-2) = 3 \). For \( h(x) = -2 \), we look for the \( x \)-value where the \( y \)-coordinate is -2. From the graph, the point with \( y = -2 \) is at \( x = -1 \) (since the minimum is at \( x = -1 \), \( y = -2 \)). So:

Step1: Find \( h(-2) \)

To find \( h(-2) \), locate \( x = -2 \) on the \( x \)-axis, then find the corresponding \( y \)-value (the height of the graph at \( x = -2 \)). From the graph, when \( x = -2 \), the \( y \)-value is 3. So \( h(-2) = 3 \).

Step2: Find \( x \) where \( h(x) = -2 \)

To find an \( x \) such that \( h(x) = -2 \), locate \( y = -2 \) on the \( y \)-axis, then find the corresponding \( x \)-value (the horizontal position of the graph at \( y = -2 \)). From the graph, the point with \( y = -2 \) has an \( x \)-value of -1 (since the minimum point is at \( x = -1 \), \( y = -2 \)). So one value of \( x \) is -1.

Answer:

(a) \( h(-2) = \boxed{3} \)
(b) One value of \( x \) for which \( h(x) = -2 \) : \(\boxed{-1}\)