Bijection/Examples/ax+b on Real Numbers

Example of Bijection

Let $a, b \in \R$ such that $a \ne 0$.

Let $f_{a, b}: \R \to \R$ be the mapping defined on the set of real numbers as:

$\forall x \in \R: \map f x = a x + b$

Then $f$ is a bijection.

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

Then:

$\ds \map f {x_1}$

$=$

$\ds \map f {x_2}$

by supposition

$\ds \leadsto \ \ $

$\ds a x_1 + b$

$=$

$\ds a x_2 + b$

Definition of $f$

$\ds \leadsto \ \ $

$\ds x_1$

$=$

$\ds x_2$

Hence by definition $f$ is an injection.

$\Box$

Let $y \in \R$.

Let $x = \dfrac {y - b} a$.

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.

$\blacksquare$