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.

Answers (1)

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.

 

 

 

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