Question

determine the range of the following graph:
(graph is shown with x-axis from -12 to 12 and y-axis from -12 to 12, with a piecewise function graph. answer section has attempt 1 out of 2, a text box, and buttons for symbols like ∅, or, ∪, ∞, <, >, ≤, ≥, ,, ,), (,, (,), and all real numbers)

Explanation:

Step1: Identify the lowest y - value

Looking at the graph, the lowest point on the graph (the minimum y - value) is at \(y = 2\). We can see from the graph that there is a part of the curve that reaches down to \(y = 2\) (the vertex of the right - hand parabola - like part) and the left - hand point also has a y - value that is higher than or equal to 2.

Step2: Identify the highest y - value

The highest point on the graph (the maximum y - value) is at \(y = 7\). We can see from the graph that the peak of the middle curve and the right - hand endpoint both have a y - value of 7 or lower? Wait, no. Wait, the left - hand dot is at \(x=-5\), let's check its y - value. From the grid, the left - hand dot is at \(y = 2\)? Wait, no, wait the left - hand dot: looking at the y - axis, the left - hand dot (at \(x = - 5\)) has a y - value of 2? Wait, no, the right - hand dot is at \(x = 3\), \(y=7\). The middle peak: let's see, the middle curve goes up to \(y = 7\) as well? Wait, no, the left - hand dot: let's check the coordinates. The left - hand dot is at \((-5,2)\)? Wait, no, the vertical axis (y - axis) has grid lines. Let's re - examine: the bottom of the right - hand parabola is at \(y = 2\) (between \(x = 0\) and \(x = 2\)). The left - hand dot is at \(x=-5\), and its y - value is 2? Wait, no, the middle curve: from \(x=-5\) (the left - hand dot) to \(x=-3\) or so, the curve goes up to a peak, then down, then the right - hand parabola. Wait, the left - hand dot is at \((-5,2)\)? Wait, no, the y - coordinate of the left - hand dot: looking at the y - axis, the left - hand dot is at \(y = 2\)? Wait, no, the right - hand dot is at \((3,7)\). The minimum y - value on the graph: the lowest point is \(y = 2\) (the bottom of the right - hand parabola and the left - hand dot's y - value). The maximum y - value: the peak of the middle curve and the right - hand dot's y - value is 7. Wait, the right - hand dot is at \(y = 7\), and the middle curve's peak is also at \(y = 7\)? Wait, no, the middle curve: from \(x=-5\) (y = 2) up to a peak, then down to \(y = 4\) or so, then the right - hand parabola goes up to \(y = 7\) at \(x = 3\). Wait, I think I made a mistake. Let's look again:

The graph has three parts: the left - hand segment (from \((-5,2)\) to the peak of the middle curve), the middle curve (from the left - hand dot to the point where it meets the right - hand parabola), and the right - hand parabola (from the intersection point to \((3,7)\)).

Wait, the range of a function is the set of all possible y - values. So we need to find the minimum and maximum y - values.

Looking at the graph:

- The lowest y - value: the bottom of the right - hand parabola is at \(y = 2\), and the left - hand dot is also at \(y = 2\). So the minimum y - value is 2.

- The highest y - value: the right - hand dot is at \(y = 7\), and the peak of the middle curve is also at \(y = 7\) (or maybe slightly less, but from the graph, the right - hand dot is at \(y = 7\), so the maximum y - value is 7.

Wait, no, the left - hand dot: let's check the coordinates. The left - hand dot is at \(x=-5\), and its y - coordinate is 2 (since it's on the same horizontal line as the bottom of the right - hand parabola). The right - hand dot is at \(x = 3\), y - coordinate 7. The middle curve: from \(x=-5\) (y = 2) up to a peak, then down, then the right - hand parabola goes from \(y = 2\) (at \(x = 0\) or so) up to \(y = 7\) at \(x = 3\). So the range is all y - values from 2 to 7, inclusive.

So the range is \(2\leq y\leq7\), or in interval notation \([2,7]\).

Wait, let's confirm:

- The minimum y - value: the lowest point on the graph is at \(y = 2\) (both the left - hand dot and the bottom of the right - hand parabola are at \(y = 2\)).

- The maximum y - value: the highest point on the graph is at \(y = 7\) (the right - hand dot and the peak of the middle curve are at \(y = 7\)).

So the range is all real numbers y such that \(2\leq y\leq7\), which is \([2,7]\).

Answer:

\([2,7]\)