Inverse Mapping/Examples/x^2-4x+5

Examples of Inverse Mappings
Let $f: \R \to \R$ be the real function defined as:


 * $\forall x \in \R: f \paren x = x^2 - 4 x + 5$

Consider the following bijective restrictions of $f$:

The inverse of $f_1$ is:


 * $\forall y \in \hointr 1 \to: \map {f_1^{-1} } y = 2 - \sqrt {y - 1}$

The inverse of $f_2$ is:


 * $\forall y \in \hointr 1 \to: \map {f_2^{-1} } y = 2 + \sqrt {y - 1}$

Proof
Let $y = \map f x$.

Then:

$f_1$ is the bijective restriction of $f$ where $x \le 2$.

Hence the negative square root is taken of $\sqrt {y - 1}$, and so:


 * $\map {f_1^{-1} } y = 2 - \sqrt {y - 1}$

$f_2$ is the bijective restriction of $f$ where $x \le 2$.

Hence the positive square root is taken of $\sqrt {y - 1}$, and so:


 * $\map {f_2^{-1} } y = 2 + \sqrt {y - 1}$