Final answer:
The question deals with the range of a mathematical function and its implications for the function's output values. It also touches on the concept of uniform distribution in the context of probability. The answer explains how the range corresponds to the set of all possible output values and how to find the probability for a specific interval within a uniform distribution.
Step-by-step explanation:
The question concerns the range of the function f(x), specifically f(x) = -2(6x) + 3, and how it relates to the concept of the function's output values for a given domain. To understand this, we first note that the range is the set of all possible output values (y-values) the function can produce given its domain (the set of all possible input values or x-values). Since the range given is (-∞, 3], this implies that for the values of x within the function's domain, f(x) will never exceed 3 but can take on any value less than or equal to 3.
Understanding Range and Domain
In the context of probability and statistics, when one says that the value of x is just as likely to be any number between two values, this is generally referring to a uniform distribution. If the function represents a continuous probability density function (PDF), and the function is constant within a domain, say 0 ≤ x ≤ 12, then the probability that x falls within the entire domain is 1, since it is certain that x will be within that range.
Probability in Uniform Distribution
For functions representing a uniform probability distribution, such as f(x) = 1/8 for 0 ≤ x ≤ 8, to find P(a < x < b) where a and b are within the domain, you calculate the area under the graph of the function between x = a and x = b.