# Show that f : R → R, given by

Question:

Show that $f: R \rightarrow R$, given by $f(x)=x-[x]$, is neither one-one nor onto.

Solution:

We have, $f(x)=x-[x]$

Injection test:

$f(x)=0$ for all $x \in \mathbf{Z}$

So, f is a many-one function.

Surjection test:

Range $(f)=[0,1) \neq \mathbf{R}$.

So, f is an into function.

Therefore, f is neither one-one nor onto.