Final answer:
The student's question pertains to the understanding of piecewise functions in mathematics, particularly graphing different expressions based on intervals of a variable and finding probabilities using a cumulative distribution function.
Step-by-step explanation:
Understanding Piecewise Functions in Mathematics
The function described in the student's question is a piecewise function, which is defined by different formulas for different intervals of the independent variable, in this case, x. Since the function's definition changes based on x being less than or greater than the variable c, you have two separate expressions to consider for x.
To graph this function, you would start by finding the value of c, which serves as the boundary between the two function expressions. Graphing the two separate expressions will involve plotting the points and drawing the curve or line according to the given function for the specified range of x.
For instance, if x is a real number within the range of 0 to 20, and the graph of f(x) is a horizontal line, this usually means that the function has a constant value within that interval.
On the other hand, if we have to find the probability P(2.5 < x < 7.5) for a continuous distribution or calculate cumulative probabilities, we would use the given function's cumulative distribution function (CDF). In the context of a function with a positive value and positive decreasing slope at x = 3, options like y = x² correctly represent the scenario as the slope of the tangent to the curve decreases as x increases.