Inverse Mapping/Examples

Examples of Inverse Mappings

$x^3$ Function on Real Numbers

Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:

$\forall x \in \R: \map f x = x^3$

The inverse of $f$ is:

$\forall y \in \R: \inv f y = \sqrt [3] y$

Bijective Restrictions of $f \paren x = x^2 - 4 x + 5$

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 $f$:

\(\ds f_1: \hointl \gets 2\)

\(\to\)

\(\ds \hointr 1 \to\)

\(\ds f_2: \hointr 2 \to\)

\(\to\)

\(\ds \hointr 1 \to\)

The inverse of $f_1$ is:

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

The inverse of $f_2$ is:

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

Arbitrary Finite Set with Itself

Let $X = Y = \set {a, b}$.

Consider the mappings from $X$ to $Y$:

\(\text {(1)}: \quad\)

\(\ds \map {f_1} a\)

\(=\)

\(\ds a\)

\(\ds \map {f_1} b\)

\(=\)

\(\ds b\)

\(\text {(2)}: \quad\)

\(\ds \map {f_2} a\)

\(=\)

\(\ds a\)

\(\ds \map {f_2} b\)

\(=\)

\(\ds a\)

\(\text {(3)}: \quad\)

\(\ds \map {f_3} a\)

\(=\)

\(\ds b\)

\(\ds \map {f_3} b\)

\(=\)

\(\ds b\)

\(\text {(4)}: \quad\)

\(\ds \map {f_4} a\)

\(=\)

\(\ds b\)

\(\ds \map {f_4} b\)

\(=\)

\(\ds a\)

We have that:

$f_1$ is the inverse mapping of itself

$f_4$ is the inverse mapping of itself

the inverse of neither $f_2$ nor $f_3$ are mappings.