Final answer:
The question delves into the realm of continuous probability distributions and seeks to explore properties like how the total area under the probability density function (PDF) curve equals one, and how probabilities are determined by areas under this curve.
Step-by-step explanation:
The student's question refers to a continuous random variable and its associated probability density function (PDF). A PDF describes the probabilities for continuous random variables by the area under the curve of the function. The total area under this curve must equal one to represent the certainty that the random variable will take on a value within the possible range. It's also noted that the area under the curve between two values gives the probability that the random variable falls within that range.
To define the subject of this question, we would typically need specific information about the function f(x) or the random variable X. However, based on the content provided, we can infer that the question is about understanding and calculating probabilities for a continuous random variable using the PDF. Tasks such as defining the random variable, graphing the probability distribution, and understanding the cumulative distribution function (cdf) are subtopics under the broader mathematical subject of probability and statistics.