What is the range of $f(x)=2 \sqrt{-x}+2$?
A. $(-\infty, 2)$
B. $[2, \infty)$
C. $(-\infty, 2]$
D. $(2, \infty)$
Answer (1)
Determine the domain of the function: x ≤ 0 .
Analyze the behavior of the function: as x goes from − ∞ to 0 , − x goes from ∞ to 0 .
Determine the range of 2 − x : it goes from ∞ to 0 .
Determine the range of f ( x ) = 2 − x + 2 : it goes from 2 to ∞ , so the range is [ 2 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 − x + 2 and asked to find its range. The range is the set of all possible output values of the function.
Finding the Domain First, we need to determine the domain of the function. Since we have a square root, the expression inside the square root must be non-negative. Thus, − x ≥ 0 , which means x ≤ 0 . So, the domain of the function is ( − ∞ , 0 ] .
Analyzing the Function Now, let's analyze the behavior of the function within its domain. As x varies from − ∞ to 0 , − x varies from ∞ to 0 . Therefore, − x varies from ∞ to 0 .
Determining the Range Since − x varies from ∞ to 0 , 2 − x also varies from ∞ to 0 . Finally, 2 − x + 2 varies from ∞ to 2 .
Final Answer Thus, the range of the function f ( x ) = 2 − x + 2 is ( 2 , ∞ ) .
Conclusion Therefore, the range of the function is [ 2 , ∞ ) .
Examples
Consider a scenario where you are tracking the distance a toy car moves from a starting point. The distance can be modeled by the function f ( x ) = 2 − x + 2 , where x represents time (but since time cannot be negative in this context, we use − x to ensure the value inside the square root is non-negative). The range of this function tells you the possible distances the car can reach. Understanding the range helps you set realistic expectations for how far the car can travel, which is useful for designing experiments or predicting performance.
