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Chapter 8

Rational Functions

A rational function is special in that the function rule involves a fraction with a polynomial in both the numerator and the denominator. Rational functions have restrictions in their domain; any value creating a 0 in the denominator has to be excluded. Many of these exclusions are identified as vertical asymptotes. The x-intercepts of rational functions can be solved for by setting the numerator equal to 0; this is done after you’ve determined that there are no common factors in the numerator and denominator. A rational function can have a horizontal asymptote — as long as the highest power in the numerator is not greater than that in the denominator.

The Problems You’ll Work On

In this chapter, you’ll work with rational functions in the following ways:

 Determining the domain and range of the function

 Removing discontinuities when possible

 Finding limits at infinity and infinite limits

 Writing equations of vertical, horizontal, and slant asymptotes

 Solving for intercepts

 Graphing rational functions

What to Watch Out For

Don’t let common mistakes trip you up; watch out for the following ones when working with rational functions:

 Categorizing a discontinuity as a vertical asymptote rather than removable

 Not dividing correctly when solving for the horizontal asymptote

 Sketching the curve on the wrong side of the horizontal asymptote

Investigating the Domain of a Rational Function

411–420 Determine the domain of the rational function.

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Determining a Function’s Removable Discontinuity

421–430 Find the removable discontinuity of the function.

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Evaluating a Function When x Is Infinitely Large or Small

431–440 Find the limit of the function as x approaches ∞.

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Determining a Function’s Infinite Limit

441–450 Find the indicated limit.

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Writing Equations of Asymptotes

451–460 Identify all vertical and horizontal asymptotes of the rational function.

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Adding Oblique Asymptotes to the Mix

461–465 Identify all vertical and oblique asymptotes.

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Sketching the Graphs of Rational Functions

466–480 Sketch the graph of the rational function, indicating all intercepts and asymptotes.

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