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Reviewed CSCA Mathematics question · Hard

What is the range of the function f(x) = 3 - 2√(4 - x^2)?

  1. [-1, 3]
  2. [-1, 7]
  3. [3, 7]
  4. [-5, 3]
Show the answer and explanation

Correct answer

A. [-1, 3]

Principle or equation

The expression 4 - x^2 is nonnegative for x in [-2,2], and √(4 - x^2) ranges from 0 to 2. Thus 2√(4 - x^2) ranges from 0 to 4, so 3 - 2√(4 - x^2) ranges from 3 - 4 = -1 to 3 - 0 = 3.

Why this answer is correct

The domain is [-2,2] because 4 - x^2 ≥ 0. The minimum of √(4 - x^2) is 0 when x = ±2, giving f = 3. The maximum is 2 when x = 0, giving f = 3 - 4 = -1. Since the square root is continuous, the range is [-1,3].

Example

For f(x) = 5 - √(9 - x^2), the range is [2,5] because √(9 - x^2) ∈ [0,3].

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