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

Which of the following functions from R to R is both one-one (injective) and onto (surjective)?

The function f(x) = 3x − 5 is both one-one and onto R. Different inputs give different outputs, and every real y has a real pre-image x = (y + 5)/3. The other options treat x and −x alike and miss negative values, so they fail.

  1. Af(x) = x²
  2. Bf(x) = 3x − 5Correct
  3. Cf(x) = |x|
  4. Df(x) = x² + 1

Explanation

f(x) = 3x − 5 is linear with non-zero slope. If 3a − 5 = 3b − 5 then a = b, so it is one-one. For any y, x = (y + 5)/3 is a real number, so it is onto. x², |x| and x² + 1 give the same value for x and −x, so they are not one-one; also none of them takes negative values (x² + 1 takes only values ≥ 1), so they are not onto R.

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