**Question:**

Give examples of two one-one functions $f_{1}$ and $f_{2}$ from $R$ to $R$, such that $f_{1}+f_{2}: R \rightarrow R$. defined by $\left(f_{1}+f_{2}\right)(x)=f_{1}(x)+f_{2}(x)$ is not one-one.

**Solution:**

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

Proving $f_{1}$ is one-one:

Let $f_{1}(x)=f_{1}(y)$

$\Rightarrow x=y$

So, $f_{1}$ is one-one.

Proving $f_{2}$ is one-one:

Let $f_{2}(x)=f_{2}(y)$

$\Rightarrow-x=-y$

$\Rightarrow x=y$

So, $f_{2}$ is one-one.

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

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 ever number in the domain is same as 0.

Thus, $\left(f_{1}+f_{2}\right)$ is not one-one.