Question

The function h(x) is defined as shown. What is the range of h(x)?

a) All real numbers
b) h(x) ≥ 0
c) h(x) ≤ 0
d) 0 < h(x) < 1

Answer 1

To determine the range of the function h(x), we need to examine the possible values that h(x) can take.

Looking at the options given:

a) All real numbers: This means that h(x) can take any value on the number line, including positive, negative, and zero.

b) h(x) ≥ 0: This means that h(x) can take any value that is greater than or equal to zero. In other words, h(x) can be zero or any positive value.

c) h(x) ≤ 0: This means that h(x) can take any value that is less than or equal to zero. In other words, h(x) can be zero or any negative value.

d) 0 < h(x) < 1: This means that h(x) can take any value between zero and one, but it cannot be zero or one itself.

Based on these options, the correct answer is a) All real numbers. The range of h(x) includes all possible real numbers, from negative infinity to positive infinity. This means that h(x) can be any positive, negative, or zero value.

I hope this clarifies the concept for you! If you have any further questions, feel free to ask.

Answer 2

The range of h(x) in the context of a continuous probability distribution and given the information provided, would most appropriately be 0 < h(x) < 1, assuming h(x) represents a non-zero, non-one constant probability.

Step-by-step explanation:

The function h(x) described appears to be part of a continuous probability distribution. The range of a function is the set of all possible output values (the y-values). For probability functions, the range is indeed constrained by the principles of probabilities, whereby any probability value must be between 0 and 1, including 0 and 1 themselves. So, if we are given that the function is a horizontal line represented by f(x), and x is restricted between 0 and 20, then the range of f(x) will be a single value and will reflect the constant height of the continuous probability distribution over the interval [0, 20].

Given the information about probabilities and that a horizontal line represents a specific y-value for all x in its domain, the range cannot be all real numbers, nor can it be less than zero since negative probabilities do not exist. The value of the range cannot exceed 1, as probabilities cannot be greater than 1. Therefore, without additional specific information about h(x), the most appropriate answer would be 0 < h(x) < 1, provided h(x) represents the height of a horizontal line in a uniform probability distribution, assuming it's not equal to 0 or 1. This is a typical characteristic of a uniform distribution within the given constraints.