What is the range of the function f(x) = 2 + sqrt(9 - x^2)?
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Correct answer
A. [2, 5]
Principle or equation
The range of a function is the set of all possible output values. For f(x) = 2 + sqrt(9 - x^2), the square root is defined for 9 - x^2 ≥ 0, i.e., x ∈ [-3, 3]. The square root term ranges from 0 to 3, so adding 2 gives a range from 2 to 5.
Why this answer is correct
Since sqrt(9 - x^2) ≥ 0 and its maximum is 3 when x = 0, the minimum of f is 2 + 0 = 2 and the maximum is 2 + 3 = 5. The function is continuous, so all values between 2 and 5 are attained. Thus the range is [2, 5].
Example
For f(x) = 1 + sqrt(4 - x^2), the range is [1, 3] because sqrt ranges from 0 to 2.
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