Prove that the function
Question:

Prove that the function $f: R \rightarrow R: f(x)=2 x$ is one-one and onto.

 

Solution:

To prove: function is one-one and onto

Given: $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}: \mathrm{f}(\mathrm{x})=2 \mathrm{x}$

We have,

$f(x)=2 x$

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

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

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

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

∴ f(x) is one-one

$f(x)=2 x$

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

$\Rightarrow y=2 x$

$\Rightarrow x=\frac{y}{2}$

Since $y \in R$,

$\Rightarrow \frac{y}{2} \in R$

⇒ x will also be a real number, which means that every value of y is associated with some x

$\therefore f(x)$ is onto

Hence Proved

 

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