Prove that the function


Prove that the function $f: N \rightarrow N: f(x)=3 x$ is one-one and into.


To prove: function is one-one and into

Given: $f: N \rightarrow N: f(x)=3 x$

We have,

$f(x)=3 x$

For, $f\left(x_{1}\right)=f\left(x_{2}\right)$

$\Rightarrow 3 x_{1}=3 x_{2}$

$\Rightarrow x_{1}=x_{2}$

When, $f\left(x_{1}\right)=f\left(x_{2}\right)$ then $x_{1}=x_{2}$

$\therefore f(x)$ is one-one

$f(x)=3 x$

Let $f(x)=y$ such that $y \in N$

$\Rightarrow y=3 x$

$\Rightarrow x=\frac{y}{3}$

If $v=1$

$\Rightarrow x=\frac{1}{3}$

But as per question $x \in N$, hence $x$ can not be $\frac{1}{3}$

Hence $f(x)$ is into

Hence Proved


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