Write whether f : R → R, given by
Question:

Write whether $f: R \rightarrow R$, given by $f(x)=x+\sqrt{x^{2}}$, is one-one, many-one, onto or into.

Solution:

$f(x)=x+\sqrt{x^{2}}=x \pm x=0$ or $2 x$

So, each element $x$ in the domain may contain 2 images.

For example,

$f(0)=0+\sqrt{0^{2}}=0$

$f(-1)=-1+\sqrt{(-1)^{2}}=-1+\sqrt{1}=-1+1=0$

Here, the image of 0 and $-1$ is 0 .

Hence, $f$ is may-one.