Final answer:
To determine the value of a piecewise function such as f(x) = x³ for a specific x, raise x to the third power. The behavior of the function is increasing within any interval where x is positive. In the context of probability distribution functions, probabilities for exact values in continuous distributions are 0, and the area under the curve represents probability.
Step-by-step explanation:
When evaluating the piecewise function f(x) = x³, the value of f(x) for a specific x value is calculated by simply raising x to the third power. If we look at the behavior of f(x) in different intervals, such as for 0 ≤ x ≤ 20, we see that f(x) will increase as x increases since the function x³ is always increasing for positive values of x.
For questions involving continuous probability distribution functions, the concepts of area under the curve and probabilities are fundamental. For instance, the probability P(x > 15) for a function defined within 0 ≤ x ≤ 15 is 0, since there are no values greater than 15. Similarly, P(x = 7) is also 0 in a continuous distribution, because the probability of exactly one value in a continuous set is always 0.
To graph a continuous probability function, you would typically plot the function as provided and shade the relevant area to find the probability for the given interval. The area under the curve in this context represents the total probability for the distribution, which should always equal 1. From the given list, the function y = x² could represent f(x) at x = 3 because the curve for x² has a positive slope that is decreasing in magnitude as x increases.