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

(N/A) To prove that $f$ is one-one,we assume $f(x_1) = f(x_2)$ for any $x_1, x_2 \in R$.
$2x_1 = 2x_2$
Dividing both sides by $2$,we get $x_1 = x_2$.
Since $f(x_1) = f(x_2)$ implies $x_1 = x_2$,the function $f$ is one-one.
To prove that $f$ is onto,we take any element $y \in R$ (codomain).
We need to find an $x \in R$ (domain) such that $f(x) = y$.
$2x = y \Rightarrow x = \frac{y}{2}$.
Since $y \in R$,$\frac{y}{2}$ is also a real number,so $\frac{y}{2} \in R$.
For every $y \in R$,there exists $x = \frac{y}{2} \in R$ such that $f(x) = f(\frac{y}{2}) = 2(\frac{y}{2}) = y$.
Thus,$f$ is onto.