Given examples of two functions f : N → N and g : N → N such that g o f is onto but f is not onto. (Hint: Consider f ( x ) = x + 1 and
It is given that the function f ( x ) is defined in the domain f : N → N and the function g ( x ) is defined in the domain g : N → N .
Consider f ( x ) = x + 1 and g ( x ) = { x − 1 if x > 1 1 if x = 1 } .
First, show that the function g ( x ) is not onto.
Consider an element 1 that exists in the co-domain N . Also, this element is not the image of any element in the domain N .
Thus, the function f is not onto.
Now, find the value of g o f ( x ) .
g o f ( x ) = g ( f ( x ) ) = g ( x + 1 ) = ( x + 1 ) − 1 = x
It can be observed that for y ∈ N , there exist x = y ∈ N such that g o f ( x ) = y .
Hence, it is proved that g o f is onto, but f is not onto.
It is given that the function
Consider
First, show that the function
Consider an element
Thus, the function
Now, find the value of
It can be observed that for
Hence, it is proved that
