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# 7: Rational Functions

• • Contributed by David Arnold
• Retired Professor (Mathematics) at College of the Redwoods
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• 7.1: Introducing Rational Functions
In this section, our study will lead us to the rational functions. Note the root word “ratio” in the term “rational.” Does it remind you of the word “fraction”? It should, as rational functions are functions in a very specific fractional form.
• 7.2: Reducing Rational Functions
The goal of this section is to learn how to reduce a rational expression to “lowest terms.” Of course, that means that we will have to understand what is meant by the phrase “lowest terms.” With that thought in mind, we begin with a discussion of the greatest common divisor of a pair of integers.
• 7.3: Graphing Rational Functions
We’ve seen that the denominator of a rational function is never allowed to equal zero; division by zero is not defined. So, with rational functions, there are special values of the independent variable that are of particular importance. Now, it comes as no surprise that near values that make the denominator zero, rational functions exhibit special behavior, but here, we will also see that values that make the numerator zero sometimes create additional special behavior in rational functions.
• 7.4: Products and Quotients of Rational Functions
In this section we deal with products and quotients of rational expressions.
• 7.5: Sums and Differences of Rational Functions
In this section we concentrate on finding sums and differences of rational expressions.
• 7.6: Complex Fractions
In this section we learn how to simplify what are called complex fractions that both the numerator and denominator are fraction problems in their own right, lending credence to why we refer to such a structure as a “complex fraction.”
• 7.7: Solving Rational Equations
When simplifying complex fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate expression could “clear” all fractions from the numerator and denominator, greatly simplifying the rational expression. In this section, a similar technique is used.
• 7.8: Applications of Rational Functions
In this section, we will investigate the use of rational functions in several applications.