First slide

Functions (XII)

Question

$Let f : N ⟶ Z and f (x) =$ $= [\begin{array}{ll} \frac{x - 1}{2}; when x is odd \\ - \frac{x}{2}; when x is even \end{array}, then :$

Moderate

A

$f (x) is bijective$
B

$f (x) is injective but not surjective$
C

$f (x) is not injective but surjective$
D

$f (x) is neither injective nor surjective$

Solution

$f (x) = [\begin{array}{cc} \frac{x - 1}{2} & x = odd \\ - \frac{x}{2} & x = even \end{array} f (x) : N \to Z$
$Let x = odd = (2 n + 1); n > 0$
$f (x) = \frac{2 n + 1 - 1}{2} = n \Rightarrow + ve integer$
$Let x = even = 2 m; m > 0$
$f (x) = - \frac{2 m}{2} = - m \Rightarrow - ve integer$
$\Rightarrow Range = codomains \Rightarrow onto and clearly f (x) is one-one function.$
Hence, bijective

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