# Give examples of two surjective functions

Question:

Give examples of two surjective functions $f_{1}$ and $f_{2}$ from $Z$ to $Z$ such that $f_{1}+f_{2}$ is not surjective.

Solution:

We know that $f_{1}: R \rightarrow R$, given by $f_{1}(x)=x$, and $f_{2}(x)=-x$ are surjective functions.

Proving $f_{1}$ is surjective :

Let $y$ be an element in the co-domain $(R)$, such that $f_{1}(x)=y$.

$f_{1}(x)=y$

$\Rightarrow x=y$, which is in $R$.

So, for every element in the co-domain, there exists some pre-image in the domain.

So, $f_{1}$ is surjective.

Proving $f_{2}$ is surjective :

Let $y$ be an element in the co domain $(R)$ such that $f_{2}(x)=y$.'

$f_{2}(x)=y$

$\Rightarrow x=y$, which is in $R .$

$\Rightarrow x=y$, which is in $R$.

So, for every element in the co-domain, there exists some pre-image in the domain.

So, $f_{2}$ is surjective.

Proving $\left(f_{1}+f_{2}\right)$ is not surjective :

Given:

$\left(f_{1}+f_{2}\right)(x)=f_{1}(x)+f_{2}(x)=x+(-x)=0$

So, for every real number $x,\left(f_{1}+f_{2}\right)(x)=0$

So, the image of every number in the domain is same as 0 .

$\Rightarrow$ Range $=\{0\}$

Co-domain $=R$

So, both are not same.

So, $f_{1}+f_{2}$ is not surjective.