What is the range of the function f(x) = 2 - sqrt(4 - x^2)?
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Correct answer
A. [0, 2]
Principle or equation
The range of a function is the set of all possible output values. For f(x) = 2 - sqrt(4 - x^2), the square root term ranges from 0 to 2, so f(x) ranges from 2 - 2 = 0 to 2 - 0 = 2.
Why this answer is correct
The expression under the square root, 4 - x^2, must be nonnegative, so x ∈ [-2, 2]. The square root sqrt(4 - x^2) takes values from 0 (when x = ±2) to 2 (when x = 0). Therefore, f(x) = 2 - sqrt(...) takes values from 2 - 2 = 0 to 2 - 0 = 2. Since the square root is continuous, all values in [0,2] are attained.
Example
For f(x) = 1 - sqrt(9 - x^2), the range is [1-3, 1-0] = [-2, 1].
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