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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

Multiply the functions.

Substitute for f(x) and g(x).

Multiply.

Combine like terms.

So, (f · g)(x) = 8x3 + 43x2 - 30x.

 (f · g)(x) = f(x) · g(x)

= (8x - 5) · (x2 + 6x)

= 8x3 + 48x2 - 5x2 - 30x

= 8x3 + 43x2 - 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) = 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.

 
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