Show that the function $f: R \rightarrow R : f(x) = 1 + x^2$ is many - one into.

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Show that the function $f: R \rightarrow R : f(x) = 1 + x^2$ is many – one into.

Show that the function $f: R \rightarrow R : f(x) = 1 + x^2$ is many – one into.

Solution:

We need to show that $f: R \rightarrow R$ given by $f(x) = 1 + x^2$ is many-one into.
A function which is not onto is into.
A function where more than one element in Set A maps to one
element in Set B is many-one.
$f(x) = 1 + x^2$
For $x = 1$ , $f(x) = 2$
For $x = -1$ , $f(x) = 2$
$\therefore f(x) = 1 + x^2$ is many-one.

$\mathrm{f}(\mathrm{x})=1+\mathrm{x}_{2}$
As the range of $\mathrm{f}(\mathrm{x})$ is $[1, \infty]$ is not equal to the codomain which is the set of Real numbers.
Therefore $\mathrm{f}(\mathrm{x})=1+\mathrm{x} 2$ is not onto, thus it is into.
As a result, $\mathrm{f}(\mathrm{x})$ is many-one into.