Question

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

Answer 1

Define $f:N\to Zasf\left(x\right)=x$ and $g:Z\to Zasg\left(x\right)=|x|.$
We first show that g is not injective. It can be observed that
g(-1)=|-1|=1, g(1)=|1|=1

Therefore, g(-1)=g(1)but $-1\ne 1$. Therefore, g is not injective.
Now, $gof:N\to Z$ is defined as gof(x)=g(f(x))=g(x)=|x|
Let $x,y\in N$ such that $gof\left(x\right)=gof\left(y\right)\Rightarrow |x|=|y|$
Since, x and $y\in N$, both are positive.
$\therefore |x|=|y|\Rightarrow x=y$
Hence, gof is injective.