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Reviewed CSCA Mathematics question · Hard

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

  1. [2, 4]
  2. [0, 2]
  3. [2, 6]
  4. [0, 4]
Show the answer and explanation

Correct answer

A. [2, 4]

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 to 4.

Why this answer is correct

Since 4 - x^2 ≥ 0, we have -2 ≤ x ≤ 2. The minimum of sqrt(4 - x^2) is 0 (at x = ±2), and the maximum is 2 (at x = 0). Therefore, f(x) = 2 + sqrt(4 - x^2) has minimum 2 and maximum 4, so the range is [2, 4].

Example

For f(x) = 1 + sqrt(9 - x^2), the range is [1, 4] because sqrt(9 - x^2) ranges from 0 to 3.

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