Question

Which of the following functions from $Z$ into $Z$ are bijective?

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

Answer 1

Answer: (B)

$f(x)=x^3$ cannot be onto as range of
$f=\{...$, $-27$, $-8$, $-1$, $0$, $1$, $8$, $27$, $...\}\ne Z$
$f(x)=2x+1$ is also not onto as
$R_f=\{...$, $-3$, $-1$, $1$, $3$, $...\}\ne Z$
$f(x)=x^2+1$ is not one-one as $f(x)=f(-x)=x^2+1$
And $f(x)=x+2$ is one-one as $f(x_1)=f(x_2)$
$\Rightarrow x_1=x_2$
and it is onto also $[\because R_f=Z\}$
Hence, $f(x)=(x+2)$ is bijective.