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