Inverse of Linear Function on Real Numbers

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Theorem

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

Let $f: \R \to \R$ be the real function defined as:

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


Then the inverse of $f$ is given by:

$\forall y \in \R: \map {f^{-1} } y = \dfrac {y - b} a$


Proof

We have that Linear Function on Real Numbers is Bijection.

Let $y = \map f x$.

Then:

\(\ds y\) \(=\) \(\ds \map f x\)
\(\ds \) \(=\) \(\ds a x + b\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \dfrac {y - b} a\)

and so:

$\forall y \in \R: \map {f^{-1} } y = \dfrac {y - b} a$

$\blacksquare$


Sources