# Bijection/Examples/2x+1 Function on Real Numbers

## Example of Bijection

Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:

$\forall x \in \R: \map f x = 2 x + 1$

Then $f$ is a bijection.

## Proof

Let $x_1$ and $x_2$ be real numbers.

Then:

 $\ds \map f {x_1}$ $=$ $\ds \map f {x_2}$ by supposition $\ds \leadsto \ \$ $\ds 2 x_1 + 1$ $=$ $\ds 2 x_2 + 1$ Definition of $f$ $\ds \leadsto \ \$ $\ds x_1$ $=$ $\ds x_2$

Hence by definition $f$ is an injection.

$\Box$

Let $y \in \R$.

Let $x = \dfrac {y - 1} 2$.

We have that:

$x \in \R$
$\map f x = y$

Hence by definition $f$ is a surjection.

$\Box$

Thus $f$ is both an injection and a surjection, and so a bijection

$\blacksquare$