Q

# Give examples of two functions f defined from N to N and g defined fromGive examples of two functions f : N → N and g : N → N such that g o f is onto but f is not onto. N to N such that g o f is onto but f is not onto.

Q. 7 Give examples of two functions $f : N\rightarrow N$ and $g : N\rightarrow N$ such that $gof$ is onto
but $f$ is not onto.

(Hint : Consider $f(x) = x + 1$ and $g(x) = \left\{\begin{matrix} x -1 & if x > 1\\ 1 & if x = 1 \end{matrix}\right.$

Views

$f : N\rightarrow N$        and         $g : N\rightarrow N$

$f(x) = x + 1$     and        $g(x) = \left\{\begin{matrix} x -1 & if x > 1\\ 1 & if x = 1 \end{matrix}\right.$

Onto :

$f(x) = x + 1$

Consider element in codomain N . It is clear that this element is not an image of any of element  in domain N .

$\therefore$    f is not onto.

$gof : N\rightarrow N$

$gof(x)= g(f(x))= g(x+1)= x+1-1 =x \, \, \, \, \, \, \, \, \, since\, x \in N\Rightarrow x+1> 1$

Now, it is clear that  $y \in N$ , there exists  $x=y \in N$     such that  $gof(x)=y$.

Hence, $gof$ is onto.

Exams
Articles
Questions