Question

Which of the following functions from Z to Z is both one-one and onto ?

• A. $f(x) = 2x - 1$
• B. $f(x) = 3x^2 + 5$
• C. $f(x) = x + 5$
• D. $f(x) = 5x^3$

Answer 1

Answer: (C)

$f(x) = x + 5$

Answer 2

A function $f: \mathbb{Z} \to \mathbb{Z}$ is one-one if $f(x_1) = f(x_2) \implies x_1 = x_2$, and onto if for every $y \in \mathbb{Z}$, there exists an $x \in \mathbb{Z}$ such that $f(x) = y$.

• $f(x) = 2x - 1$: One-one, but not onto (e.g., $y=0$ has no integer solution for $x$).
• $f(x) = 3x^2 + 5$: Not one-one ($f(2) = f(-2)$) and not onto.
• $f(x) = x + 5$: One-one and onto (for any $y \in \mathbb{Z}$, $x = y-5 \in \mathbb{Z}$ such that $f(x) = y$).
• $f(x) = 5x^3$: One-one, but not onto (e.g., $y=1$ has no integer solution for $x$).