Composition of Mappings is not Commutative/Examples/Arbitrary Example 1
Jump to navigation
Jump to search
Example of Compositions of Mappings
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = x^2 + 1$
Let $g: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map g x = x + 1$
Then the compositions of $f$ with $g$ are:
\(\ds f \circ g: \R \to \R: \, \) | \(\ds \map {\paren {f \circ g} } x\) | \(=\) | \(\ds \paren {x + 1}^2 + 1\) | \(\ds = x^2 + 2 x + 2\) | ||||||||||
\(\ds g \circ f: \R \to \R: \, \) | \(\ds \map {\paren {g \circ f} } x\) | \(=\) | \(\ds \paren {x^2 + 1} + 1\) | \(\ds = x^2 + 2\) |
and it is immediately seen that:
- $g \circ f \ne f \circ g$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): composite function (function of a function)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): composite function (function of a function)