Show that the function $f : R \rightarrow R : f(x) = x^4$ is many - one and into.

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Show that the function $f : R \rightarrow R : f(x) = x^4$ is many – one and into.

Show that the function $f : R \rightarrow R : f(x) = x^4$ is many – one and into.

Solution:

We need to show that f: $\mathrm{R} \rightarrow \mathrm{R}$ given by $\mathrm{f}(\mathrm{x})=\mathrm{x} 4$ 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.
$f(x)=x^{4}$
For $x=1, f(x)=1$
For $x = -1$ , $f(x) = 1$
$\therefore f(x) = x^4$ is many-one.

$\mathrm{f}(\mathrm{x})=\mathrm{x} 4$
As the range of $\mathrm{f}(\mathrm{x})$ is $[0, \infty]$ is not equal to the codomain which is the set of Real numbers.
Therefore $\mathrm{f}(\mathrm{x})=\mathrm{x} 4$ is not onto, thus it is into.
As a result, $f(x)$ is many-one into.