Question

find the domain and range of the function:
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, it appears to be a parabola (a quadratic function) that extends horizontally across all real numbers? Wait, no, wait. Wait, looking at the graph, the points are at x = - 3, - 2, - 1, 0, 1, 2, 3? Wait, no, the graph has points at x=-3 (y = 18), x=-2 (y = 8), x=-1 (y = 2), x = 0 (y = 0), x = 1 (y = 2), x = 2 (y = 8), x = 3 (y = 18). Wait, maybe it's a function defined for x from - 3 to 3? Wait, no, the arrows at the top? Wait, no, the graph is a parabola - like shape with points at x=-3, -2, -1, 0, 1, 2, 3. Wait, maybe the domain is all real numbers from - 3 to 3? Wait, no, the graph's x - axis has markings from - 5 to 5, but the function's graph is between x=-3 and x = 3? Wait, no, the points are at x=-3, -2, -1, 0, 1, 2, 3. Wait, maybe the domain is \(\{-3, - 2, - 1, 0, 1, 2, 3\}\)? Wait, no, maybe I misread. Wait, the graph is a parabola (quadratic function) with vertex at (0,0) and passing through (-1,2), (-2,8), (-3,18), (1,2), (2,8), (3,18). Wait, actually, the function seems to be \(y = 2x^{2}\) when x is - 1,1 (y = 2), x=-2,2 (y = 8), x=-3,3 (y = 18). Wait, but the domain: looking at the graph, the x - values for which the function is defined. The graph has points at x=-3, -2, -1, 0, 1, 2, 3. So the domain is the set of x - values from - 3 to 3, inclusive. So domain is \([-3, 3]\) (if it's a continuous function? Wait, no, the points are discrete? Wait, no, the graph is a smooth curve? Wait, the image shows a curve with points, but maybe it's a function defined for x in \(\{-3, - 2, - 1, 0, 1, 2, 3\}\)? Wait, no, maybe I made a mistake. Wait, let's re - examine. The graph is a parabola - like shape with vertex at (0,0) and symmetric about the y - axis. The x - values for which the function is defined: looking at the graph, the leftmost point is at x=-3 and the rightmost at x = 3. So the domain is all real numbers x such that \(-3\leq x\leq3\), or in set notation, \([-3, 3]\).

Step2: Determine the Range

The range of a function is the set of all possible y - values. Looking at the y - values: at x=-3, y = 18; x=-2, y = 8; x=-1, y = 2; x = 0, y = 0; x = 1, y = 2; x = 2, y = 8; x = 3, y = 18. So the y - values start from 0 (at x = 0) and go up to 18 (at x=\(\pm3\)). So the range is all real numbers y such that \(0\leq y\leq18\), or in set notation, \([0, 18]\). Wait, but let's check the minimum y - value is 0 (at x = 0) and the maximum is 18 (at x=\(\pm3\)), and all values in between (since it's a smooth curve) are included. So range is \([0, 18]\). Wait, but maybe the domain is \(\{-3, - 2, - 1, 0, 1, 2, 3\}\) and range is \(\{0, 2, 8, 18\}\)? Wait, no, the graph is a curve, so it's continuous. Wait, the points are at ( - 3,18), (-2,8), (-1,2), (0,0), (1,2), (2,8), (3,18). So the function is \(y = 2x^{2}\) for x in \(\{-3, - 2, - 1, 0, 1, 2, 3\}\)? No, that would be discrete. But the graph looks like a continuous parabola. Wait, maybe the domain is all real numbers from - 3 to 3 (inclusive) and the range is all real numbers from 0 to 18 (inclusive).

Wait, let's correct. The domain of a function is the set of all x - values for which the function is defined. From the graph, the function exists for x from - 3 to 3 (since the leftmost point is at x=-3 and the rightmost at x = 3). So domain: \(-3\leq x\leq3\) or \([-3, 3]\). The range is the set of all y - values. The minimum y - value is 0 (at x = 0) and the maximum is 18 (at x=\(\pm3\)), and all values in between are covered (since it's a parabola - like curve), so range: \(0\leq y\leq18\) or \([0, 18]\).

Answer:

Domain: \([-3, 3]\)
Range: \([0, 18]\)