Function composition
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. Function composition is represented with the "circle" operator:
- <math>(f \circ g)(x) <math>
which is equivalent to f(g(x)), where it is clear that the value of g applied to x is used as the argument for f. The notation f o g is read as "f circle g" or "f composed with g".
The composition of a function on itself, such as f o f, is customarily written f 2. Thus:
- (f o f)(x) = f(f(x)) = f 2(x)
- (f o f o f)(x) = f(f(f(x))) = f 3(x)
Note that, for historical reasons, this superscript notation represents standard exponentiation when used with trigonometric functions:
- sin2(x) = sin(x)·sin(x)
Nevertheless, an extension of this notation using negative exponents applies to all functions, including trigonometric ones:
- f -1(x) is the inverse function of f
Derivatives of compositions involving differentiable functions can always be found using the chain rule.
In some cases, an expression for f in g(x) = f r(x) can be derived from the rule for g given non-integer values of r. This is called fractional iteration.
See also
