Given examples of two functions `f:" "N ->N" "a n d""""""g:" "N->N` such that of is onto but f is not onto. (Hint: Consider `f(x)" "=" "x+1" "a n d""""""""g(x)" "=" "{x-1 if x>1 1 if x=1}

Explanation

f: `N rarr N` , is defined as f(x) = x+1 . Now
Let f(x) =1
`rArr x+ 1=1`
`rArr x=0 notin N`
`therefore ` f in not onto
Again let g: `N rarr N ` is defined as
`g(x)={{:(x-1"," if x gt 1 ),(1"," if x=1):}`
Now gof : `N rarr N` (gof)(x) = g[f(x)] = g(x + 1)
=(x+1)-1 ` ( because x +1 gt 1)`
= x For eachy `in` N, x = y `in` N is such that
(gof)(x)=y
`rArr` gof is onto