Question

Which of the following functions have a domain of (-[infinity],[infinity]) and a range of (-2,[infinity] )? Check all that apply.

1. f(x) = 0.25²- 2
2. f(x) = 3ˣ + 2
3. f(x) = 0.25ˣ + 2
4. f(x) = 3ˣ - 2

Answer 1

The two functions from the list that have a domain of (-∞,∞) and a range of (-2,∞) are f(x) = 3ˣ + 2 and f(x) = 3ˣ - 2.

Step-by-step explanation:

The student is asking which of the given functions have a domain of (-∞,∞) and a range of (-2,∞). Let's evaluate each function:

• f(x) = 0.25² - 2: This is not a function but an arithmetic expression with a constant value of -2, and thus, does not have a domain or range as it's not a function.
• f(x) = 3ˣ + 2: This exponential function's domain is all real numbers, and its range is (2,∞) because the output will always be greater than 2, regardless of the x-value. Thus, this function meets the criteria.
• f(x) = 0.25ˣ + 2: This function's domain is also all real numbers, but its range is (2,∞) since the exponential part will always be greater than or equal to 1, and adding 2 makes the range start from 2 onward.
• f(x) = 3ˣ - 2: The domain is all real numbers here as well, but the range starts at -2, which satisfies the condition of the range being (-2,∞).

Therefore, the functions that satisfy both the given domain and range are f(x) = 3ˣ + 2 and f(x) = 3ˣ - 2.