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.

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

Then:

Hence by definition $f$ is an injection.

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.

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