Final answer:
A piecewise defined function or f(x) is a function defined by different formulas depending on the interval of the input value. It is often used in probability studies, where the function f(x) is defined such that the area between it and the x-axis equals a probability, thus forming a probability density function.
Step-by-step explanation:
A piecewise defined function, commonly known as f(x), is a function that is defined by several different formulas, or 'pieces', based on where the input (x-value) lies within certain intervals. If you picture this graphically, it means that for different ranges of x, the function behaves differently. For instance, the function might be linear in one range of x-values and quadratic in another.
In the field of probability, especially with continuous probability distributions, we define the function f(x) such that the area between it and the x-axis equals a probability, hence the term probability density function (pdf). For example, in a given question, the function f could be defined as f(x) = 0.25e(-0.25x) where x is at least zero, meaning that different x-values will yield different probabilities.
In this context, piecewise defined functions provide a useful tool to represent different situations or conditions within a single mathematical construct, which is essential in understanding advanced mathematical concepts.
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