Question

this is the graph of a polynomial function. there is no absolute minimum or maximum. find the domain and range. (-∞, 0 (-∞, ∞) (-∞, ∞) 0, ∞) domain: range:

Explanation:

Step1: Determine the domain

The domain of a function is the set of all x - values for which the function is defined. Looking at the graph, the polynomial function has endpoints? No, wait, the graph extends from the left (negative x - values) to the right (positive x - values) with some endpoints? Wait, no, the left end is at x=-5 (the point there) and the right end is at x = 5? Wait, no, looking at the graph, the leftmost point is at x=-5 (the dot) and the rightmost point is at x = 5 (the dot). Wait, no, actually, the graph is a polynomial function, but the graph as drawn has a left endpoint at x=-5 and right endpoint at x = 5? Wait, no, maybe I misread. Wait, the x - axis goes from - 10 to 10, and the graph has a left endpoint at x=-5 (the bottom left dot) and right endpoint at x = 5 (the top right dot). Wait, no, the domain is the set of all x - values covered by the graph. Wait, the left end is at x=-5 (the point) and the right end is at x = 5 (the point). Wait, no, actually, looking at the x - coordinates, the graph starts at x=-5 (the leftmost point) and ends at x = 5 (the rightmost point)? Wait, no, the x - axis labels: from - 10 to 10. The left dot is at x=-5 (since the grid lines: each grid is 1 unit). Wait, the leftmost point is at x=-5 (the bottom left) and the rightmost point is at x = 5 (the top right). Wait, but the graph of a polynomial function: if it's a polynomial, the domain is usually all real numbers, but here the graph has endpoints. Wait, maybe the graph is a piecewise? No, the problem says it's a polynomial function, but the graph has two endpoints: left at x=-5 and right at x = 5? Wait, no, looking at the x - axis, the left dot is at x=-5 (the x - coordinate is - 5) and the right dot is at x = 5 (x - coordinate 5). Wait, but the domain is the set of x - values from the leftmost x to the rightmost x. Wait, the leftmost x is - 5 and the rightmost x is 5? No, wait, the x - axis: the left dot is at x=-5 (because the grid lines: from - 10, - 9,...,-5, - 4,...0,1,...5,6... So the left dot is at x=-5 (the x - coordinate) and the right dot is at x = 5 (x - coordinate). Wait, but the options for domain: one of the options is (-∞, ∞)? No, the options given are (-∞, 0], (-∞, ∞), (-∞, ∞), [0, ∞). Wait, maybe I made a mistake. Wait, the graph: looking at the x - values, the graph is defined from x=-5 to x = 5? No, the left end is at x=-5 (the bottom left) and the right end is at x = 5 (the top right). Wait, no, the x - coordinate of the leftmost point: let's count the grid lines. From x=-10 to x = 10, each grid is 1 unit. The left dot is at x=-5 (because it's 5 units to the right of - 10? No, - 10, - 9, - 8, - 7, - 6, - 5: so the left dot is at x=-5. The right dot is at x = 5 (5 units to the right of 0). Wait, but the domain of a function is all x - values for which the function exists. If the graph has endpoints at x=-5 and x = 5, then the domain is [-5,5]? But the options don't have that. Wait, the options given are (-∞, 0], (-∞, ∞), (-∞, ∞), [0, ∞). Wait, maybe the graph is actually a polynomial with domain all real numbers, but the drawing has endpoints. Wait, no, the problem says "There is no absolute minimum or maximum", which for a polynomial, if it's of odd degree, it has no absolute min or max. But the graph here has endpoints. Wait, maybe the graph is a cubic or something, but the options for domain: one of the options is (-∞, ∞). Wait, maybe the left and right dots are just the ends of the graph shown, but the domain is all real numbers? No, the options include (-∞, ∞) twice. Wait, maybe the domain is (-∞, ∞) because it's a polynomial function, and the graph is just a portion, but the domain of a polynomial is all real numbers. Wait, but the options: the first option is (-∞, 0], second (-∞, ∞), third (-∞, ∞), fourth [0, ∞). Wait, maybe the graph is actually defined for all real numbers, so domain is (-∞, ∞).

Step2: Determine the range

The range is the set of all y - values. Looking at the graph, the bottom point is at y=-10 (at x=-5) and the top point is at y = 10 (at x = 5)? No, the left dot (x=-5) has y=-10, and the right dot (x = 5) has y = 10. Wait, but the graph in between: when x is from - 5 to 5, what's the range? Wait, the graph goes from y=-10 (at x=-5) up to y = 10 (at x = 5), but also, looking at the middle, when x = 0, y = 0. Wait, but the options for range: one of the options is (-∞, ∞)? No, the options for range: let's see the options. Wait, the options given for the four boxes: first (-∞, 0], second (-∞, ∞), third (-∞, ∞), fourth [0, ∞). Wait, maybe the range is all real numbers? No, the graph has a minimum at y=-10 and maximum at y = 10? No, the problem says "There is no absolute minimum or maximum", which contradicts. Wait, maybe I misread the graph. Wait, the graph: on the left side, it goes from x=-5 (y=-10) up to x = 0 (y = 0), and on the right side, from x = 0 (y = 0) up to x = 5 (y = 10). Wait, no, the left side: from x=-5 (y=-10) to x = 0 (y = 0), and right side: from x = 0 (y = 0) to x = 5 (y = 10). So the range is from y=-10 to y = 10? But the options don't have that. Wait, the options for domain and range: the first box is (-∞, 0], second (-∞, ∞), third (-∞, ∞), fourth [0, ∞). Wait, maybe the domain is (-∞, ∞) (since it's a polynomial, even though the graph has endpoints, maybe it's a mistake, and the domain is all real numbers) and the range is (-∞, ∞)? No, the graph has a minimum at y=-10 and maximum at y = 10? But the problem says "There is no absolute minimum or maximum", which means the range is all real numbers. Wait, maybe the graph is a cubic function, which has no absolute min or max, so the range is (-∞, ∞). And the domain: for a polynomial, domain is (-∞, ∞). Wait, the options: two of the options are (-∞, ∞) for domain and range? Wait, the problem has four boxes: first (-∞, 0], second (-∞, ∞), third (-∞, ∞), fourth [0, ∞). So Domain: let's see, the graph: does it cover all x - values? The left end is at x=-5, but maybe it's a mistake, and the domain is (-∞, ∞) (since it's a polynomial). Range: since there's no absolute min or max, the range is (-∞, ∞).

Wait, maybe the graph is a polynomial of odd degree, so domain is all real numbers (-∞, ∞) and range is all real numbers (-∞, ∞). So Domain: (-∞, ∞), Range: (-∞, ∞).

Answer:

Domain: \((-\infty, \infty)\)
Range: \((-\infty, \infty)\)