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 Dependent Variable

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# 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) = x2 + 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) = 8x3 + 43x2 - 30x. (f Â· g)(x) = f(x) Â· g(x) = (8x - 5) Â· (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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