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Thus one obtains a function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. The notation g o f is read as "g circle f" or "g composed with f".
As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x). Then (c o h)(t) describes the oxygen concentration around the plane at time t.
In the mid- 20th century, some mathematicians decided that writing "g of" to mean "first apply f, then apply g" was too confusing and decided to change notations. They wrote "xf" for "f(x)" and "xfg" for "g(f(x))". However, this movement never caught on, and nowadays this notation is found only in old books.
The functions g and f are commutative if g o f = f o g.
Derivatives of compositions involving differentiable functions can be found using the chain rule. See also Faà di Bruno's formula.If Y⊂X then f may compose with itself; this is sometimes denoted f 2. Thus:
The functional powers f of n = f n o f = f n+1 for
naturaln follow immediately.
Do not confuse it with the notation commonly seen in
trigonometry in which, 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:
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 .