Give examples of two functions

Let $f: N \rightarrow Z$ be given by $f(x)=x$, which is injective.

(If we take f(x) = f(y), then it gives x = y)

Let $g: Z \rightarrow Z$ be given by $g(x)=|x|$, which is not injective.

If we take $f(x)=f(y)$, we get:

$|x|=|y|$

$\Rightarrow x=\pm y$

Now, gof: $N \rightarrow Z$.

$(g \circ f)(x)=g(f(x))=g(x)=|x|$

Let us take two elements x and y in the domain of gof , such that

$(g \circ f)(x)=(g \circ f)(y)$

$\Rightarrow|x|=|y|$

$\Rightarrow x=y(\mathrm{We}$ don't get $\pm$ here because $\mathrm{x}, \mathrm{y} \in \mathrm{N})$

So, gof is injective.