Is the function f: r to r defined by f(x) =x^2+1 for 'x' belongs to real number (r) an into function? give reason.

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 is an into function if

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 , if is another real number?

The answer is yes: the equation represents a parabola with vertex , so the graph never goes below 1. In other words, the range of 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.