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Chapter 4: Problem 4

Where are the vertical asymptotes of a rational function located?

Short Answer

Expert verified
Question: Explain the process of finding the vertical asymptotes of a rational function R(x) = P(x)/Q(x). Answer: To find the vertical asymptotes of a rational function, follow these steps: 1. Identify the denominator Q(x) of the given rational function. 2. Find the zeros of the denominator Q(x) by setting it to zero and solving for x. 3. Check the numerator P(x) at each of the zeros found for the denominator. If P(x) is not equal to zero, a vertical asymptote exists at that point. 4. The vertical asymptote(s) can be written as x = a, x = b, etc., where a and b are the zeros of the denominator Q(x) that fulfill the condition mentioned in step 3.

Step by step solution

01

Understand the definition of a rational function

A rational function is any function that can be expressed as the quotient of two polynomial functions, P(x) and Q(x). The rational function is represented as R(x) = P(x)/Q(x). The vertical asymptotes occur where the denominator Q(x) is equal to zero, and the numerator P(x) is not equal to zero, which causes the function R(x) to approach infinity or negative infinity.
02

Identify the denominator Q(x)

Given the rational function R(x) = P(x)/Q(x), first identify the denominator Q(x). This is the polynomial function in the bottom of the fraction.
03

Find the zeros of the denominator Q(x)

In order to find the vertical asymptotes, you need to determine the zeros of the denominator Q(x). To do this, set the denominator Q(x) equal to zero and solve for x: Q(x) = 0. The zeros of the denominator will represent the x-values where the function approaches infinity or negative infinity.
04

Check the numerator P(x)

For each zero of the denominator found in step 3, examine the corresponding value of the numerator P(x). If P(x) is not equal to zero at any of the zeros found in step 3, then a vertical asymptote exists at that location.
05

Identify the vertical asymptotes

Once you have checked the numerator P(x) for each of the zeros of the denominator Q(x) and confirmed that P(x) is not equal to zero at those points, the vertical asymptote will occur at those x-values. The vertical asymptotes can be written as x = a, x = b, and so on, where a and b are the zeros of the denominator Q(x) that fulfill the condition provided. By following these steps, you can find the vertical asymptotes of any given rational function R(x) = P(x)/Q(x).

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