The idea behind composing functions is to take the function of another function. Many of the models you many see in the life sciences are the composition of simpler functions.
Definition
Let f(x) and g(x) be two functions.
Definition: We define the composition of f and g, denoted
by,
where the domain of
is all x in the domain of g such that such that g(x) is in the domain of f. |
The notation f (g(x)) is read as, “f of g of x.”
Breaking down composite functions
To break down the composite function,
we work from the inside out. First, we take a value of x in the domain of g and evaluate g(x). Next, we take the value of g(x) that we just got, and substitute it into f(x). But, we can only do this if g(x) is in the domain of f. We can represent this with a diagram,
We will now look at some examples of composite functions. First, consider the functions,
f(x) = 3x − 1,
g(x) = x2.
Using the definition of composition, we find
as follows,
The domain of
consists of all x in the domain of g such that g(x) is in the domain of f . To determine the domain, we start with the inside function and work outward. The inside function is g(x), and the domain of g(x) is all real numbers. Now we need to be sure that the value of g(x) (i.e. the range of g) is in the domain of f (x). Any value of g(x), however, is in the domain of f (x) since f has domain all real numbers. Therefore, the domain of
is all real numbers.
In a similar manner, we can also find
as,
For similar reasons as above, the domain of
is also all real numbers.
A note of caution
Notice that is generally not equal to .
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Now consider the following functions,
f (x) = 1 − 8x,
g(x) = 2 / x .
We find
as follows,
Thus, we find
The domain of
consists of all x in the domain of g such that g(x) is in the domain of f . We determine the domain by starting with the inside function and working outward. The inside function is g(x), and the domain of g(x) is {x | x ≠ 0}. Now we need to be sure that the value of g(x) (i.e. the range of g) is in the domain of f(x). Any value of g(x), however, is in the domain of f(x) because f has domain all real numbers.
Therefore the domain of
is {x | x ≠ 0} (i.e. all x in the domain of g such that g(x) is the domain of f).
In a similar manner, we can also find
as,
To find the domain
we need to find all x in the domain of f such that f (x) is in the domain of g. The domain of f is all real numbers. The domain of g is {x | x ≠ 1/8 }. The range of f is all real numbers, therefore f can take on the
value 1/8 and thus, we must remove it from the domain. Therefore, we conclude that the domain of
is {x | x ≠ 1/8} .
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In the next section we explore inverse functions.
Inverse Functions |