Product and Quotient of Functions

We can also form new functions by multiplying or dividing.

Definition — Product and Quotient of Two Functions

Given two functions, f(x) and g(x):

The product of f and g, written (f · g)(x), is defined as (f · g)(x) = f(x) · g(x).

The domain of (f · 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) = x² + 6x, find the product (f · g)(x).

Solution

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) = x² - 11x + 30 and g(x) = x² - 25, find the quotient

Solution

Divide the functions.
Substitute for f(x) and g(x).
To reduce the fraction, first factor.
Now, cancel common factors of x - 5.

So,

Note:

To factor x² - 11x + 30, find two integers whose product is 30 and whose sum is -11. They are -5 and -6.

To factor x² - 25, find two integers whose product is -25 and whose sum is 0. They are -5 and 5.