College Algebra 9th txtbk

226 CHAPTER 3 FUNCTIONS
EXAMPLE 2 Finding the Quotient of Two Functions
Let f
and g(x) x 4 Find the function and find its domain.
x 1
x 3 .
g
SOLUTION
Because division by 0 must be excluded, the domain of f is all x except x 1 and the domain
of g is all x except x 3. Now we find fg.
a f f (x)
b(x)
g g (x)
x
x 1
x 4
x 3
x
x 1 x 3
x 4
x(x 3)
(x 1)(x 4)
(1)
The fraction in equation (1) indicates that 1 and 4 must be excluded from the domain of
fg to avoid division by 0. But equation (1) does not indicate that 3 must be excluded
also. Although the fraction in equation (1) is defined at x 3, 3 was excluded from
the domain of g, so it must be excluded from the domain of fg also. The domain of fg
is all real numbers x except 3, 1, and 4.
MATCHED PROBLEM 2
f
Let f (x) 1 and g (x) x 5 . Find the function and find its domain.
x 2
x
g
Z Composition
Consider the functions f and g given by
f (x) 1x
and
g(x) 4 2x
Note that g(0) 4 2(0) 4 and f(4) 14 2. So if we apply these two functions
consecutively, we get
f (g(0)) f (4) 2
In a diagram, this would look like
x 0
g(x)
4
f(x)
2
When two functions are applied consecutively, we call the result the composition of functions.
We will use the symbol f g to represent the composition of f and g, which we
formally define now.