Product and Quotient of Functions
We can also form new functions by multiplying or dividing.
Definition â€” Product and Quotient of Two Functions
Given two functions, f(x) and g(x):
The product of f and g, written (f Â· g)(x), is defined as
(f Â· g)(x) = f(x)
Â· g(x).
The domain of (f Â· g)(x) consists of all real numbers that are in the
domain of both f(x) and g(x).
The quotient of f and g, written
, is defined as
Here g(x)
≠
0.
The domain of
consists of all real numbers that
are in the domain of both f(x) and g(x) and for which g(x)
≠ 0.
Example 1
Given f(x) = 8x  5 and g(x) = x^{2} + 6x, find the product (f
Â· g)(x).
Solution
Multiply the functions. Substitute for f(x) and g(x).
Multiply.
Combine like terms.
So, (f Â· g)(x) = 8x^{3} +
43x^{2}  30x. 
(f Â· g)(x) 
= f(x) Â· g(x)
= (8x  5) Â· (x^{2} + 6x)
= 8x^{3} + 48x^{2}  5x^{2}  30x
= 8x^{3} + 43x^{2}  30x 
To multiply two binomials, use FOIL
(First, Outer, Inner, Last).
That is,
(a + b)(c + d) = ac + ad + bc + bd.
Example 2
Given f(x) = x^{2}  11x + 30 and g(x) = x^{2}  25, find the quotient
Solution
Note:
To factor x^{2}  11x + 30, find two integers
whose product is 30 and whose sum is 11. They are 5 and 6.
To factor x^{2}  25, find two integers whose
product is 25 and whose sum is 0. They
are 5 and 5.
