Question

determine the range of the following graph:
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∅ or ∪ ∞ < > ≤ ≥ , (,) (, (,)
all real numbers

Explanation:

Step1: Identify the lowest y - value

The graph has a minimum point at \(y=-1\) (the vertex at the origin - like point on the y - axis? Wait, looking at the graph, the leftmost point is at \(x = - 9,y=-1\)? Wait, no, the graph touches the y - axis at \(y=-1\)? Wait, actually, looking at the graph, the lowest point (the minimum y - value) is \(y=-1\) (from the point where the graph touches the y - axis or the lowest part). Wait, no, the left endpoint is at \((-9, - 1)\)? Wait, no, the blue dot on the left is at \(x=-9,y = - 1\)? Wait, the graph: let's check the y - coordinates. The leftmost point (blue dot) is at \(y=-1\), the middle part goes up, and the rightmost blue dot is at \(y = 5\) (x = 3, y = 5). Wait, actually, the range is the set of all y - values the graph takes. The lowest y - value is \(-1\) (since the graph has a point at \(y=-1\) and goes up from there, and also the middle part goes up to some maximum, but wait, the left endpoint is at \(y=-1\), the graph then rises, falls, and then rises again to \(y = 5\) at \(x = 3\). Wait, no, let's re - examine. The graph starts at \((-9,-1)\) (blue dot), goes up, then down, then up to \((3,5)\) (blue dot). So the minimum y - value is \(-1\) and the maximum y - value is 5? Wait, no, when the graph goes up from \((-9,-1)\), it reaches a peak, then comes down to cross the x - axis, then goes down to \(y=-1\) at \(x = 0\)? Wait, maybe I misread. Wait, the graph: the left blue dot is at \((-9,-1)\), then the graph curves up, then down, crosses the x - axis, then curves up again, with a blue dot at \((3,5)\). Wait, the lowest y - value is \(-1\) (since the graph has points with \(y=-1\) and all y - values between \(-1\) and 5, because the graph goes from \(y=-1\) up to 5, and includes all values in between. So the range is all real numbers \(y\) such that \(-1\leq y\leq5\)? Wait, no, let's check the y - coordinates. The left point: \(y=-1\), the right point: \(y = 5\), and the graph is continuous between them, so the range is \([-1,5]\). Wait, let's confirm: the range of a function is the set of all output values (y - values) it can take. The graph starts at \(y=-1\) (left endpoint), goes up, then down, then up to \(y = 5\) (right endpoint). So the minimum y is \(-1\) and the maximum y is 5, and since the graph is continuous, all y - values between \(-1\) and 5 (inclusive) are included.

Step2: Determine the range

The range is the interval of y - values. The smallest y - value (minimum) is \(-1\) and the largest y - value (maximum) is \(5\), and since the graph includes these endpoints (the blue dots are closed points), the range is \([-1,5]\).

Answer:

\([-1, 5]\)