Chapter 7 Rational Functions - College of the Redwoods

Section 7.8 Applications of Rational Functions 731
7.8 Applications of Rational Functions
In this section, we will investigate the use of rational functions in several applications.
Number Problems
We start by recalling the definition of the reciprocal of a number.
Definition 1. For any nonzero real number a, the reciprocal of a is the number
1/a. Note that the product of a number and its reciprocal is always equal to the
number 1. That is,
a · 1
a = 1.
For example, the reciprocal of the number 3 is 1/3. Note that we simply “invert”
the number 3 to obtain its reciprocal 1/3. Further, note that the product of 3 and its
reciprocal 1/3 is
3 · 1
3 = 1.
As a second example, to find the reciprocal of −3/5, we could make the calculation
(
1
− 3 = 1 ÷ − 3 ) (
= 1 · − 5 )
= − 5 5
3 3 ,
5
but it’s probably faster to simply “invert” −3/5 to obtain its reciprocal −5/3. Again,
note that the product of −3/5 and its reciprocal −5/3 is
(
− 3 ) (
· − 5 )
= 1.
5 3
Let’s look at some applications that involve the reciprocals of numbers.
◮ Example 2.
The sum of a number and its reciprocal is 29/10. Find the number(s).
Let x represent a nonzero number. The reciprocal of x is 1/x. Hence, the sum of x
and its reciprocal is represented by the rational expression x + 1/x. Set this equal to
29/10.
x + 1 x = 29
10
To clear fractions from this equation, multiply both sides by the common denominator
10x.
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Version: Fall 2007