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Reviewed CSCA Mathematics question · Hard

What is the range of the function f(x) = 2 / (x^2 + 4x + 5)?

  1. (0, 2]
  2. (0, 2)
  3. [0, 2]
  4. (0, ∞)
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Correct answer

A. (0, 2]

Principle or equation

The range of a rational function with a quadratic denominator can be found by analyzing the minimum or maximum of the denominator.

Why this answer is correct

The denominator is x^2 + 4x + 5 = (x + 2)^2 + 1, which has a minimum value of 1 at x = -2. Thus the maximum value of f is 2/1 = 2, and as the denominator grows without bound, f approaches 0 but never reaches it. Hence the range is (0, 2].

Example

For f(x) = 1/(x^2 + 1), the range is (0, 1].

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