Prove that the function `f: N->N` , defined by `f(x)=x^2+x+1` is one-one but not onto.

Explanation

Let x.y `in` N and f(x) = f(y)
`rArr x^(2)+x+1=y^(2)+y+1`
`rArr x^(2)-y^(2)+x-y=0`
`rArr (x-y)(x+y)+(x-y) =0`
`rArr(x-y)(x+y+1)=0`
`rArr x-y=0`
`rArr` x=y
Therefore f is one-one.
Again, let f (a) = b, where b `in` N
`rArr a^(2)+a+1 =b`
`rArr a^(2)+a+(1-b)=0`
`rArra = (-1+-sqrt(1-4(1-b)))/(2)`
`rArra = (-1+-sqrt(4b-3))/(2)in N, if b = 2 in N`
Therefore , 2 is an element in co-domain N such that it has no pre-image in domain N.
Therefore, f is not onto. Hence Proved.