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).
To determine the valid prediction about the continuous function f(x), we need to understand what a continuous function is and how to analyze its behavior over different intervals.
A continuous function is a function that has no breaks, holes, or jumps in its graph. This means that the graph of a continuous function is a connected curve without any interruptions.
Now let's analyze the given answer choices:
A). f(x) ≤ 0 over the interval (–∞, ∞):
This answer choice states that the function is less than or equal to 0 for all real numbers. Since a continuous function can take on any value, it is possible for it to be below, above, or equal to 0. Therefore, this prediction is valid.
B). f(x) > 0 over the interval (–1, ∞):
This answer choice states that the function is greater than 0 for all x-values greater than -1. A continuous function can have portions above or below 0, so this prediction may or may not be valid.
C). f(x) ≥ 0 over the interval [–1, 1]:
This answer choice states that the function is greater than or equal to 0 for all x-values between -1 and 1, inclusive. A continuous function can have portions below 0, so this prediction may or may not be valid.
D). f(x) < 0 over the interval (0, 2):
This answer choice states that the function is less than 0 for all x-values between 0 and 2. A continuous function can have portions above 0, so this prediction may or may not be valid.
To make a valid prediction about the function f(x), we would need more information about its behavior, such as a specific equation or a graph. Without additional information, we cannot definitively determine which prediction is valid.