Question

determine the range of the following graph:

Explanation:

Step1: Identify the minimum y - value

The graph has a lowest point (from the visible part and the endpoints). The left - most point is an open circle, but we look at the actual plotted part. The lowest y - value we can see from the graph (considering the curve and the endpoints) is - 4 (from the left - hand curve part) and the right - hand endpoint has a y - value that is above - 4? Wait, no. Wait, the left - hand part: the curve goes down to a minimum (the open circle is at x=-9, but the curve below? Wait, no, let's re - examine. The graph: the left - hand part has a curve that goes down, then up, crosses the x - axis at x=-3, then goes up to a maximum, then down to the right - hand endpoint at x = 9 (with a closed dot). Wait, the y - values: the minimum y - value on the graph (the lowest point of the curve) is - 4 (from the left - hand curve segment) and the maximum y - value is 6 (at the peak of the parabola - like part). Wait, let's check the y - axis. The left - hand curve: when x=-9, there is an open circle, but the curve below? Wait, no, the graph: the left - hand part (from x < - 9? No, the x - axis is labeled from - 12 to 12. The graph has a curve that starts from an open circle at x=-9 (so x=-9 is not included) and goes down? Wait, no, the curve: from the open circle at x=-9, it goes down to a minimum, then up, crosses the x - axis at x=-3, then goes up to a maximum (around x = 4, y = 6), then down to the closed dot at x = 9, y=-4? Wait, no, the right - hand endpoint is at x = 9, and looking at the y - axis, the dot is at y=-4? Wait, no, the y - axis: the numbers are 12,11,10,9,8,7,6,5,4,3,2,1,0,-1,-2,-3,-4,-5,... So the left - hand curve: the minimum y - value (the lowest point of the function's graph, considering the domain) is - 4 (since the left - hand part's curve goes down to y=-4, and the right - hand endpoint is at y=-4? Wait, no, the right - hand endpoint is at x = 9, and the y - value there is - 4? Wait, the peak is at y = 6 (around x = 4). So the range is the set of all y - values that the graph takes. The minimum y - value is - 4 (inclusive? Wait, the left - hand open circle is at x=-9, but the curve below that? No, the open circle is at x=-9, so x=-9 is not in the domain, but the y - values of the graph: the lowest y - value is - 4 (from the curve) and the highest is 6. Wait, let's confirm:

• The maximum y - value: the graph reaches up to y = 6 (at the vertex of the upper parabola - like part).
• The minimum y - value: the graph goes down to y=-4 (both the left - hand curve segment and the right - hand endpoint? Wait, the right - hand endpoint is a closed dot, so x = 9 is included, and its y - value is - 4. The left - hand part: the curve has a minimum at y=-4 (even though x=-9 is an open circle, the y - value of - 4 is achieved at some x in the domain).

So the range is all real numbers y such that - 4 ≤ y ≤ 6. Wait, let's check again. The left - hand curve: from x > - 9 (since x=-9 is open) to x=-3, it goes from y (at x > - 9) down to - 4, then up to cross the x - axis at x=-3, then up to y = 6, then down to x = 9, y=-4. So the minimum y - value is - 4 (inclusive, because the right - hand endpoint is at y=-4, and the left - hand curve also reaches y=-4 for some x in the domain) and the maximum y - value is 6 (inclusive, as the peak is achieved).

Step2: Determine the range

The range of a function is the set of all possible output (y) values. From the graph, the lowest y - value (minimum) is - 4 and the highest y - value (maximum) is 6. So the range is - 4 ≤ y ≤ 6, or in interval notation, $[-4,6]$.

Answer:

The range of the graph is $[-4, 6]$ (or all real numbers y such that $-4\leq y\leq6$)