Question

Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but g is not injective.

Answer 1

Let f : N → 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 → 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 → Z.
$\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(x\right)=\left|x\right|$
Let us take two elements x and y in the domain of gof , such that
$$\begin{gathered} \left(gof\right)\left(x\right)=\left(gof\right)\left(y\right) \\ \Rightarrow \left|x\right|=\left|y\right| \\ \Rightarrow x=y\left(\text{We don't get ± here because x, y ∈N}\right) \end{gathered}$$
So, gof is injective.