Question

Give examples of two surjective function $f_1$ and $f_2$ from $Z$ to $Z$, such that $f_1+f_2$ is not surjective.

Answer 1

A function $f:A\to B$ is said to be a onto function or surjective if every elemnt of $A$ i.e, if $f\left(A\right)=B$ or range of $f$ is the co-domain of $f.$

So, $f:A\to B$ is surjective for each $b\in B,$ there exists $a\in B$ such that $f\left(a\right)=b$

Let $f_1:Z\to Z$ and $f_2:Z\to Z$ be two functions given by

$\Rightarrow$ $f_1\left(x\right)=x$

$\Rightarrow$ $f_2\left(x\right)=-x$

From above function it is clear that both are surjective functions.

Now,

$\Rightarrow$ $f_1+f_2:Z\to Z$

$\Rightarrow$ $(f_1+f_2)\left(x\right)=f_1\left(x\right)+f_2\left(x\right)$

$\Rightarrow$ $(f_1+f_2)\left(x\right)=x-x$

$\Rightarrow$ $(f_1+f_2)\left(x\right)=0$

Therefore, $f_1+f_2:Z\to Z$ is a function given by

$\Rightarrow$ $(f_1+f_2)\left(x\right)=0$

Since, $f_1+f_2$ is a constant function, hence it is not an onto or surjective function.