Combinations of Functions
Composition of Functions - Functions of Functions
Basic functions are often combined with each other to describe more complicated situations. Here we will consider the composition of functions, functions of functions.
Definition: The composite of two functions and , written , is .
The domain of the composite function consists
of those x–values for which and are both defined - we can evaluate the composition of two functions at a point x only if each step in the composition is defined.
If
we think of our functions as machines, then composition is simply
a new machine consisting of an arrangement of the original machines. The composition of the function machines and shown in
Fig. 5(a) is an arrangement of the machines so that the original input goes into machine , the output from machine becomes the input into machine , and the output from machine is our final output.
The composition of the function machines is only valid if is an allowable input into is in the domain of and if is then an allowable input into .
The composition involves arranging the machines so the original input goes into , and the output from then becomes the input for (Fig. 5(b) ).
Example 2: For , and , evaluate , , and . Find the equations and domains of
and .
Solution:
which is undefined
.
, and the domain of is those x–values for which so the domain
of is all such that or .
, but we can evaluate the first piece, , of the composition
only if is defined, so the domain of is all .
Practice 4: For , , and .
Evaluate , , , ,
, and . Find the equations for and .