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CA Foundation · Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity

The function f(x) = x² − 4x + 7 is defined on the set of real numbers. What is its range?

The range is [3, ∞). Writing f(x) as (x − 2)² + 3 shows the smallest value is 3 at x = 2, and the function grows without bound beyond that, so every value from 3 upward is attained.

  1. A[3, ∞)Correct
  2. B[7, ∞)
  3. C(−∞, 3]
  4. D[4, ∞)

Explanation

Complete the square: f(x) = (x − 2)² + 3. Since (x − 2)² ≥ 0, f(x) ≥ 3, with equality at x = 2, and f takes all values above 3. So the range is [3, ∞). The option [7, ∞) wrongly uses f(0) as the minimum.

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