Functions: Composite Functions
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.
The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. 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 ".
The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. 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 ".
The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f o (g o h) = (f o g) o h. Since there is no distinction between the choices of placement of parentheses, they may be safely left off.
Example:
For f (x) = 2x + 5, g(x) = x2, find (f o g)(x) and (g o f)(3)
Solution:
(f o g)(x) = f [g(x)] = f (x2) = 2(x2) + 5 = 2x2 + 5
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