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Chapter 2: Problem 132

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

Short Answer

Expert verified
The statement 'The graph of a rational function can have three vertical asymptotes.' is true.

Step by step solution

01

Understand Vertical Asymptotes in Rational Functions

In a rational function, the vertical asymptotes occur where the denominator of the function equals zero, and the numerator does not. This is because when the denominator equals zero, the function becomes undefined at those points. The number of distinct roots (solutions) of the denominator equals the number of vertical asymptotes, assuming the numerator does not become zero at the same values.
02

Determine Condition for Three Vertical Asymptotes

A rational function can have three vertical asymptotes if the degree of the polynomial in the denominator is three and each root of the denominator's polynomial is unique. This is because each root will cause the denominator to equal zero, resulting in a vertical asymptote, and the non-zero condition of the numerator at these roots needs to be met.
03

Evaluate the initial statement

Thus, it is true to say that a rational function can have three vertical asymptotes. The condition for this is that the polynomial in the denominator of the function be of degree three (or more with repeated roots) and each root of the denominator's polynomial must be unique.

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Solve each inequality using a graphing utility. $$\frac{x-4}{x-1} \leq 0$$

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