Check whether f : N→N given by f(x) x^2 +x +1is onto.

First, let's recall what it means for a function to be
onto (also known as surjective). We say that a
function `f: X to Y` is surjective if for all
y in Y, there exists an x
in X such that y =
f(x).

Let's now apply this definition to our
function `f(x) = x^2 + x + 1` keeping in mind that the function is only defined for
natural numbers.

Any easy way to show a function is not
onto is by showing a counter example.

Suppose, for
contradiction, that f is onto. Let y = 2. Then
there exists some natural number x such that `2 = x^2 + x + 1`
Using the quadratic formula, we find that this has two
solutions:

`x= 1/2 (-1 - sqrt{5})` and `x = 1/2(sqrt{5} -
1)`

But neither of these solutions are natural numbers.
Therefore f is not
onto.