A function is said to be an into function if there are elements in the codomain that are not "reached" by any element of the domain.
In other words, a function [tex] f:A\to B [/tex] is an into function if
[tex] \exists b \in B:\ b\neq f(a) \forall a \in A [/tex]
Since the codomain of the function is the real numbers set, the question is: is there a real number that can't be written as [tex] x^2+1 [/tex], if [tex] x [/tex] is another real number?
The answer is yes: the equation represents a parabola with vertex [tex] (0,1) [/tex], so the graph never goes below 1. In other words, the range of [tex] x^2+1 [/tex] is composed by all numbers greater than or equal to 1.
So, this is an into function, because all numbers smaller than 1 belong to the codomain, but aren't reached by any number in the domain.
