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

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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.


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


\(\ds \map f {x_1}\) \(=\) \(\ds \map f {x_2}\) 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.


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.


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