Final answer:
The function f is continuous over its domain, which is a true statement. However, statements about its domain being all real numbers and its specific value at f(1) are unverifiable or false based on the given information.
Step-by-step explanation:
Considering the information provided, let's address the truthfulness of each statement about the function f:
- A. As x approaches positive infinity, f approaches positive infinity.
This statement is not provided in the information and cannot be verified. - B. The function f is continuous over its domain.
This is true because, by definition, a continuous probability density function is continuous over its defined range. This is supported by the given info where f(x) is defined as a continuous probability function. - C. The domain of function f is all real numbers.
This is false, as the function is restricted within a range, such as 0 ≤ x ≤ 20 or 1 ≤ x ≤ 4 based on the provided examples, and not all real numbers. - D. The value of f(1) is 2.
The value of f(x) at any specific x is not given in the provided information, leaving this statement unverifiable. - E. Function f is decreasing over the entire domain.
Without a specific function or additional information about its slope we cannot conclude this.
Based on the information about continuous probability functions and the given examples, we can also address some basic concepts of probability:
- The probability of an event where x is outside of its defined range is zero, as in P(x > 15) when 0 ≤ x ≤ 15.
- For a continuous distribution, the probability of a specific value, such as P(x = 7), is zero because the probability is calculated over intervals.
- The total area under a continuous probability density function is equal to 1 since it represents the total probability of all possible outcomes within the domain.