Understanding Rational Functions

This video tutorial explains how to determine the domain, identify holes, and find the equations of vertical and horizontal asymptotes for a given rational function. It begins with factoring the numerator and denominator, then excludes the zeros of the denominator from the domain. The video explains that there are no holes due to the absence of common factors. It identifies vertical asymptotes as the zeros of the denominator and explains the horizontal asymptote by comparing the degrees of the numerator and denominator. The video concludes with a graphical verification of these concepts.

What is the first step in analyzing a rational function?

• Identifying the holes

• Graphing the function

• Finding the horizontal asymptote

• Factoring the numerator and denominator

Why are certain x-values excluded from the domain of a rational function?

• They are not integers

• They are negative

• They are too large

• They make the function undefined

Which of the following x-values is NOT excluded from the domain of the given function?

• x = 1

• x = 0

• x = -6

• x = 4

What results in a hole in a rational function?

• A zero in both the numerator and denominator

• A zero in the denominator only

• A zero in the numerator only

• A zero in the numerator

What is the result when there are no common factors between the numerator and denominator?

• There are holes

• There are no holes

• The function is undefined

• The function has infinite solutions

What determines the vertical asymptotes of a rational function?

• Zeros of the numerator

• Degree of the denominator

• Zeros of the denominator

• Degree of the numerator

Which of the following is a vertical asymptote of the function?

• x = 0

• x = -6

• x = 2

• x = 5

How is the horizontal asymptote determined when the degree of the denominator is greater than the numerator?

• It approaches infinity

• It approaches zero

• It does not exist

• It approaches the leading coefficient

What is the horizontal asymptote of the given function?

• y = 1

• y = -1

• y = 0

• y = x

What happens to the function values as x approaches infinity?

• They increase indefinitely

• They decrease indefinitely

• They approach zero

• They remain constant