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.
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