Understanding Graphs of Rational Functions

The video tutorial explores the function f(x) = g(x)/(x^2 - x - 6), where g(x) is a polynomial. The instructor guides viewers through factoring the denominator to identify x-values that make it zero, leading to vertical asymptotes or removable discontinuities. The video evaluates multiple graph options to determine which one accurately represents the function, concluding that choice C is correct due to its vertical asymptote at x = -2 and removable discontinuity at x = 3.

What is the main focus of the problem introduced in the video?

• Solving for x in the equation

• Identifying the graph of y = f(x)

• Finding the roots of g(x)

• Determining the degree of g(x)

What is the first step in analyzing the function f(x)?

• Graphing the function

• Factoring the denominator

• Factoring the numerator

• Finding the roots of g(x)

What are the critical x-values for the denominator of f(x)?

• x = -3 and x = 2

• x = 1 and x = -1

• x = 3 and x = -2

• x = 0 and x = 1

What does a zero denominator indicate in the context of the graph?

• A vertical asymptote or removable discontinuity

• A horizontal asymptote

• A local maximum or minimum

• A point of inflection

What condition must be met for a removable discontinuity to occur?

• The numerator must not be zero at the same x-value

• The denominator must be zero at a different x-value

• The function must be continuous

• The numerator must be zero at the same x-value

Which graph choice has a vertical asymptote at x = -2?

• Choice B

• Choice A

• Choice D

• Choice C

Why is Choice A ruled out as a possible graph for f(x)?

• It has no vertical asymptotes

• It has a vertical asymptote at x = 3

• It is defined at x = 3

• It has a removable discontinuity at x = -2

What is the issue with Choice B as a graph for f(x)?

• It has no vertical asymptotes

• It has a removable discontinuity at x = -2

• It has a vertical asymptote at x = 3

• It is defined at x = 3

Which graph choice correctly represents a removable discontinuity at x = 3?

• Choice D

• Choice C

• Choice B

• Choice A

Why is Choice D not a valid graph for f(x)?

• It has no vertical asymptotes

• It has vertical asymptotes at incorrect x-values

• It is defined at x = -2

• It has a removable discontinuity at x = 3