Question

1. what are the domain and range? a domain: all real numbers range: y ≥ -8 b domain: x ≤ 4 range: -8 ≤ y ≤ 8 c domain: x ≤ 4 range: y ≥ -8 d domain: x ≥ 4 range: -8 ≤ y ≤ 8

Explanation:

Step1: Analyze the graph's domain

The graph appears to be a parabola (or a similar curve) that extends infinitely in the horizontal direction (left and right) without any breaks or restrictions. So the domain should be all real numbers, as there's no limit on the x - values the graph can take.

Step2: Analyze the graph's range

Looking at the graph, the lowest point (minimum value of y) is at \(y=- 8\), and the graph extends upwards from there (since it's a parabola - like shape opening upwards). So the range should be all real numbers \(y\) such that \(y\geq - 8\).

Now let's check the options:

• Option A: Domain: all real numbers, Range: \(y\geq - 8\) matches our analysis.
• Option B: The domain \(x\leq4\) is incorrect as the graph extends beyond \(x = 4\) (and to the left without a limit at \(x = 4\)), and the range \(-8\leq y\leq8\) is incorrect as the graph goes above \(y = 8\) (the point at \(x = 4\) has \(y = 8\) but the graph is increasing, so \(y\) can be greater than 8? Wait, no, looking at the graph, the upper point is at \(y = 8\) but actually, the curve is a parabola? Wait, no, maybe it's a piece - wise? Wait, no, the left part is a parabola opening upwards with vertex at some point with \(y=-8\), and the right part is a line? Wait, no, maybe it's a quadratic function. Wait, the key is: for domain, since the graph has no restrictions on x (it's defined for all x - values, as it's a continuous curve from left to right), domain is all real numbers. For range, the minimum y - value is \(-8\) (the lowest point of the parabola - like part) and then it goes up, so \(y\geq - 8\). Option A has domain all real numbers and range \(y\geq - 8\). Option B: domain \(x\leq4\) is wrong, range \(-8\leq y\leq8\) is wrong because the right part is increasing, so y can be greater than 8? Wait, no, the point on the right is at \(x = 4\), \(y = 8\), and then it's a line going up? Wait, maybe the graph is a quadratic function. Wait, maybe I misread. Let's re - check: the left part is a parabola opening upwards with vertex at \(y=-8\), and the right part is a line? No, maybe it's a single function. Anyway, the correct domain is all real numbers (since there's no restriction on x) and range \(y\geq - 8\) (since the lowest y is \(-8\) and it goes up from there). So option A is correct.

Answer:

A. Domain: all real numbers; Range: \(y\geq - 8\)