Prove that the function $f: R \rightarrow R$, given by $f(x)=2 x,$ is one-one and onto.
Prove that the function $f: R \rightarrow R$, given by $f(x)=2 x,$ is one-one and onto.
$f$ is one-one, as $f\left(x_{1}\right)$ $=f\left(x_{2}\right) \Rightarrow 2 x_{1}$ $=2 x_{2} \Rightarrow x_{1}=x_{2} .$ Also, given any real number $y$ in $R$ there exists $\frac{y}{2}$ in $R$ such that $f\left(\frac{y}{2}\right)$ $=2 \cdot\left(\frac{y}{2}\right)=y .$ Hence, $f$ is onto.
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