Which of the following functions from Z to Z are bijections ? (where Z is the set of integers)
Answer & explanation
Correct answer: option 2
Check each function for injectivity and surjectivity:
1. $f(x) = x^3$
Injective: If $x_1^3 = x_2^3 \Rightarrow x_1 = x_2$ ✅
Surjective: For any $y \in \mathbb{Z}$, $x = \sqrt[3]{y} \in \mathbb{Z}$ (only if $y$ is a perfect cube). Not all integers are perfect cubes ❌
Not bijection
2. $f(x) = x + 2$
Injective: $x_1 + 2 = x_2 + 2 \Rightarrow x_1 = x_2$ ✅
Surjective: For any $y \in \mathbb{Z}$, $x = y - 2 \in \mathbb{Z}$ ✅
Bijection ✅
3. $f(x) = x^3 + 1$
Injective: $x_1^3 + 1 = x_2^3 + 1 \Rightarrow x_1^3 = x_2^3 \Rightarrow x_1 = x_2$ ✅
Surjective: For any $y \in \mathbb{Z}$, $x^3 = y - 1$. Not all integers can be written as perfect cube + 1 ❌
Not bijection
4. $f(x) = x^2 + 1$
Injective: $x_1^2 + 1 = x_2^2 + 1 \Rightarrow x_1^2 = x_2^2 \Rightarrow x_1 = \pm x_2$ ❌
Not injective, so not bijection ❌
Answer: $f(x) = x + 2$