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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
