Give example of two functions f : N $\rightarrow$ Z and g : Z $\rightarrow$ Z such that gof is injective but g is not injective.

(Hint : Consider f(x) $=$ x and g(x) $= |x|$)

f : N $\rightarrow$ N and g : Z $\rightarrow$ Z

Let f(x) $=$ x and g(x) $=$|x|. Since, g(x)$=$g($-$x) $= |x| \forall x \in Z$

$\therefore$ g is not one$-$one $\Rightarrow$ g is not injective.

Since, f : N $\rightarrow$ Z and g : Z $\rightarrow$ Z $\Rightarrow$ gof : N $\rightarrow$ Z. Let ${x_1},{x_2} \in N.$

Now, (gof) ($x_1) = (gof) (x_2) \Rightarrow g(x_1) = g(x_2) |x_1| = |x_2|$

$\Rightarrow$ ${x_1} = {x_2}$
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$\therefore$ gof is one-one. Hence, gof is injective.