Show that the function
(i) $f: N \rightarrow N: f(x)=x^{2}$ is one - one into.
(ii) $f: Z \rightarrow Z: f(x)=x^{2}$ is many - one into
(i) $f: N \rightarrow N: f(x)=x^{2}$ is one - one into.
$f(x)=x^{2}$
$\Rightarrow y=x^{2}$
Since the function $f(x)$ is monotonically increasing from the domain $N \rightarrow N$
$\therefore f(x)$ is one -one
Range of $f(x)=(0, \infty) \neq N($ codomain $)$
$\therefore f(x)$ is into
$\therefore f: N \rightarrow N: f(x)=x^{2}$ is one - one into.
(ii) $f: Z \rightarrow Z: f(x)=x^{2}$ is many - one into
$f(x)=x^{2}$
$\Rightarrow y=x^{2}$
in this range the lines cut the curve in 2 equal valued points of $y$, therefore, the function $f(x)=x^{2}$ is many one.
Range of $f(x)=(0, \infty) \neq Z$ (codomain)
$\therefore f(x)$ is into
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.