Which of the following functions from Z to Z is a bijective function?…

The correct answer is Option (4) → \(f(x) = x+2\)

A bijective function is both one-one and onto. Each option is checked on the integers:

Option (1) \(f(x) = x^3\) — Incorrect. It is one-one, but not onto: 2 is not the cube of any integer.

Option (2) \(f(x) = 2x + 1\) — Incorrect. It is one-one, but 2x + 1 is always odd, so no even integer (for example 2) is an image. Not onto.

Option (3) \(f(x) = x^2+1\) — Incorrect. It is not one-one: f(1) = f(−1) = 2.

Option (4) \(f(x) = x+2\) — Correct.

    \(f(x_1) = f(x_2) \Rightarrow x_1 + 2 = x_2 + 2 \Rightarrow x_1 = x_2\)

So f is one-one. For any integer y, x = y − 2 is an integer with f(x) = y, so f is onto.

Tip: x³ and 2x + 1 are bijections on R, but not on Z: whether a function is onto depends on the set it maps into.