Final answer:
To extend a function and ensure it is continuous at a certain point, one must match the function values and derivatives on both sides of that point. For continuous probability density functions, probabilities are represented as areas under the curve, and probabilities of exact values are 0 in a continuous distribution.
Step-by-step explanation:
To provide a formula for an extended function that is continuous at a given point, one needs to ensure that the function value and its first derivative are the same on either side of the point. Ideally, we want to match the limits of the function as we approach our point of interest from both sides, as well as ensure that the derivative does not have a discontinuity, unless the potential function V(x) is infinite, according to the information given.
An example of a function that needs to be extended to be continuous might involve piecewise functions where different expressions are valid in different regions. In probability theory, if we are talking about a continuous probability density function, f(x), we consider the probabilities in terms of the areas under the curve of f(x). For example, the probability P(x < 0) would be zero for a continuous probability distribution that is restricted from 0 to 5 because no values below zero are included in the range.
To answer specific questions given in the context of a continuous probability density function:
- For question 6: P(x > 15) is 0, as the distribution is defined only from 0 to 15.
- For question 7: The area under f(x) must be 1 if it is a proper probability density function.
- For question 8: P(x = 7) is 0, as the probability of a precise point value in a continuous distribution is always 0.
- For question 9: P(x = 10) is also 0 since the continuous probability function does not include values beyond x = 7.
For question 11: P(0 < x < 12) is 1, assuming the continuous probability function encompasses the entire range from 0 to 12.