Question

domain :
range :

Explanation:

Step1: Determine the domain

The domain of a function is the set of all possible x - values for which the function is defined. Looking at the graph, the left - most point has an x - coordinate of - 40, and as we move along the graph, the x - values go up to 10 (since the arrow at the end of the graph is at x = 10). So the domain is all real numbers from - 40 to 10, inclusive. In interval notation, this is $[-40,10]$.

Step2: Determine the range

The range of a function is the set of all possible y - values that the function can take. Looking at the graph, the lowest point (the minimum y - value) is - 4 (from the point on the graph) and the highest point (the maximum y - value) is - 2 (from the peak of the curve). Wait, no, let's re - examine. Wait, the y - axis: the graph goes from a y - value of - 4 (at some point) up to a y - value of - 2 (at the peak). Wait, actually, looking at the grid, the left - most point has a y - coordinate of - 4 (since it's on the line y=-4), and the peak of the curve is at y = - 2. Then as x increases to 10, the y - value goes down to - 5? Wait, no, the arrow at the end (x = 10) has a y - coordinate of - 5? Wait, maybe I misread. Wait, the graph: let's check the coordinates. The left - most point is (- 40, - 4). Then the curve goes up to a maximum at y=-2 (around x=-20 to x = - 10), and then goes down. At x = 0, the y - value is - 3, and at x = 10, the y - value is - 5. So the minimum y - value is - 5 (at x = 10) and the maximum y - value is - 2 (at the peak). So the range is all real numbers from - 5 to - 2, inclusive? Wait, no, wait the left - most point is (- 40, - 4), so the y - value there is - 4. Then the peak is at y=-2, and then at x = 10, y=-5. So the range is from - 5 (the lowest y - value) to - 2 (the highest y - value)? Wait, no, because the left - most point is at y=-4, which is higher than - 5. Wait, maybe I made a mistake. Let's list the y - values: the left - most point: y=-4. The peak: y=-2. The point at x = 10: y=-5. So the set of y - values starts at - 5 (the lowest) and goes up to - 2 (the highest), because - 5 is less than - 4, so the minimum y - value is - 5 and the maximum is - 2. So the range is $[-5, - 2]$. Wait, but the left - most point is at y=-4, which is within $[-5, - 2]$ because - 5 < - 4 < - 2. Yes, because - 5 is less than - 4 and - 2 is greater than - 4. So the range is all real numbers from - 5 to - 2, inclusive.

Wait, maybe a better way: look at the vertical extent. The lowest y - value on the graph is - 5 (at x = 10) and the highest is - 2 (at the peak). So the range is $[-5, - 2]$.

Answer:

Domain: $[-40, 10]$
Range: $[-5, - 2]$