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

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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

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

Solution:

(i)

$f: N \rightarrow N: f(x)=x^{2}$ is one – one into.
As 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
$\begin{array}{l} f(x)=x^{2} \\ \Rightarrow y=x^{2} \end{array}$
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
$\therefore f : Z \rightarrow Z : f(x) = x^2$ is many – one into