Question

Consider functions f and g. f(x) = x 12/x^2 4x-12 g(x) = 4x^2-16x 16/4x 48 which expression is equal to ?

Answer 1

For a uniform probability function f(x) over the interval [0, 12], the probability P(0 < x < 12) is 1, indicating that the event x falling within this range is certain.

Step-by-step explanation:

The student seems to be working on a problem involving probability functions and perhaps is looking for the probability P(0 < x < 12) for a continuous probability function f(x). The statement 'f(x), a continuous probability function, is equal to 12, and the function is restricted to 0 ≤ x ≤ 12' indicates that f(x) is likely a uniform distribution over the interval [0, 12]. If f(x) is indeed a uniform distribution, then the probability density function would be a constant over the allowed range. Since the total area under a probability density function over its entire domain equals 1, and considering that the function f(x) is nonzero and constant between 0 and 12, we would divide 1 by the length of the interval, which is 12, to find that constant value.

The constant value of the probability density function f(x) would be 1/12. Therefore, the probability P(0 < x < 12) would simply be the area under the curve between 0 and 12, which is the product of the constance density 1/12 and the interval length, which is 12. Hence, P(0 < x < 12) = 1/12 × 12 = 1. This means that the total probability over the interval (0, 12) is 100% or certain, which makes sense for a continuous probability function over its entire domain where it is nonzero.

Answer 2

For a continuous probability function f(x) equal to 12 over the range 0 ≤ x ≤ 12, the probability P (0 < x < 12) is equal to 1. This is because excluding the endpoints from the range of a continuous function does not affect the probability.

Step-by-step explanation:

The question relates to the evaluation of a probability function within a given interval. In this case, the function f(x) represents a continuous probability function which is equal to 12 over the interval 0 ≤ x ≤ 12. To find P (0 < x < 12), we look at the properties of a probability density function over a continuous range. Since f(x) is a constant function with a value of 12, and the total area under the curve over its range (from 0 to 12) must equal 1 (since it's a probability function), we can calculate the probability for the interval 0 < x < 12 by considering the fact that the probability of the entire interval is 1.

The actual probability, P (0 < x < 12), is therefore the probability of the entire interval excluding the endpoints, which is still equal to 1, as excluding points from a continuous interval does not change the probability. Thus, P (0 < x < 12) = 1.