Absolute Value of Function is Composite with Absolute Value Function

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Theorem

Let $S$ be a set.

Let $\R$ denote the real number line.

Let $f: S \to \R$ be real-valued function.

Let $\size f$ denote the absolute value of $f$, that is, $\size f$ is the mapping defined by:

$\forall s \in S : \map {\size f} s = \size{\map f s}$

Then:

$\size f = \size{\,\cdot\,} \circ f$

where:

$\size{\,\cdot\,} : \R \to \R$ denotes the absoute value function

$\size{\,\cdot\,} \circ f$ denotes the composite mapping of $\size{\,\cdot\,}$ and $f$

Proof

We have:

\(\ds \forall s \in S: \, \)

\(\ds \map {\size f } s\)

\(=\)

\(\ds \size{\map f s}\)

Definition of Absolute Value of Real-Valued Function

\(\ds \)

\(=\)

\(\ds \map {\paren{\size{\,\cdot\,} \circ f} } s\)

Definition of Composite Mapping

By definition of equality of mappings:

$\size f = \size{\,\cdot\,} \circ f$

$\blacksquare$