Which is a valid prediction about the continuous function f(x)?
A).f(x) ≤ 0 over the interval (–∞, ∞).
B).f(x) > 0 over the interval (–1, ∞).
C).f(x) ≥ 0 over the interval [–1, 1].
D).f(x) < 0 over the interval (0, 2).
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To determine which of these predictions about the continuous function f(x) is valid, we can look at the graph provided in the link. Let's go through each option and analyze them based on the graph.
A). f(x) ≤ 0 over the interval (-∞, ∞):
Looking at the graph, we can see that there are portions of the curve that are below the x-axis (negative y-values) and other portions that are above the x-axis (positive y-values). Therefore, f(x) is not always less than or equal to 0 over the entire interval (–∞, ∞), so option A is not valid.
B). f(x) > 0 over the interval (–1, ∞):
Looking at the graph, we can see that all the y-values corresponding to x-values greater than -1 are indeed positive (greater than 0). Therefore, f(x) is greater than 0 over the interval (–1, ∞), so option B is valid.
C). f(x) ≥ 0 over the interval [–1, 1]:
Looking at the graph, we can see that all the y-values corresponding to x-values between -1 and 1 (inclusive) are either positive or zero (greater than or equal to 0). Therefore, f(x) is greater than or equal to 0 over the interval [–1, 1], so option C is valid.
D). f(x) < 0 over the interval (0, 2):
Looking at the graph, we can see that there are portions of the curve that are above the x-axis (positive y-values) and other portions that are below the x-axis (negative y-values). Therefore, f(x) is not always less than 0 over the entire interval (0, 2), so option D is not valid.
Based on the graph, the valid predictions are options B and C.