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Q.
Which of the following functions from $Z$ into $Z$ are bijective?
Relations and Functions - Part 2
Solution:
$f(x) = x^3$ 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.