Unformatted text preview: ce of ﬁnding the domain of the composite
function before simplifying. For instance, the domain of f ◦ g is much diﬀerent than its simpliﬁed
formula would indicate. Composing a function with itself, as in the case of h ◦ h, may seem odd.
Looking at this from a procedural perspective, however, this merely indicates performing a task
h and then doing it again - like setting the washing machine to do a ‘double rinse’. Composing a
function with itself is called ‘iterating’ the function, and we could easily spend an entire course on
just that. The last two problems in Example 5.1.1 serve to demonstrate the associative property
of functions. That is, when composing three (or more) functions, as long as we keep the order the
same, it doesn’t matter which two functions we compose ﬁrst. This property as well as another
important property are listed in the theorem below.
Theorem 5.1. Properties of Function Composition: Suppose f , g , and h are functions.
• h ◦ (g ◦ f ) =...
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