What is the range of the function f(x) = 5 - 2√(9 - x^2)?
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Correct answer
A. [-1, 5]
Principle or equation
The square root function √(9 - x^2) has minimum 0 and maximum 3 on its domain [-3,3]. Therefore f(x) = 5 - 2√(...) has maximum 5 - 0 = 5 and minimum 5 - 2*3 = -1. Since the square root is continuous, all values between -1 and 5 are attained.
Why this answer is correct
The expression under the square root, 9 - x^2, ranges from 0 to 9 as x varies from -3 to 3. Thus √(9 - x^2) ranges from 0 to 3. Multiplying by -2 gives values from -6 to 0. Adding 5 gives values from -1 to 5. Hence the range is [-1, 5].
Example
For f(x) = 3 - √(4 - x^2), the square root ranges from 0 to 2, so f ranges from 1 to 3. Range: [1,3].
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