How to Identify Vertical Asymptotes in Rational Functions

TL;DR

Identify vertical asymptotes and removable discontinuities in rational functions by analyzing the denominator. For the given function, the key x-values are where the denominator equals zero. Correct graph choices will reflect these asymptotic behaviors.

Transcript

• [Voiceover] We're told, let f of x equal g of x over x squared minus x minus six, where g of x is a polynomial. Which of the following is a possible graph of y equals f of x? And they give us four choices. The fourth choice is off right over here. And like always, pause the video, and see if you can figure it out or if you were having trouble wit... Read More

Key Insights

• 💁 The given function f(x) is in the form of a rational function, with a polynomial numerator and a polynomial denominator.
• ☺️ The denominator, x^2 - x - 6, can be factored into (x - 3)(x + 2), revealing the potential values of x that make the denominator equal to zero.
• ☺️ The function f(x) will have a vertical asymptote or a removable discontinuity at x = 3 or x = -2, depending on whether g(x) can be factored as (x - 3) times another polynomial.

Questions & Answers

Q: What is the given function?

The given function is f(x) = g(x) / (x^2 - x - 6), where g(x) is a polynomial.

Q: How can we determine the potential points where the denominator equals zero?

By factoring the denominator, x^2 - x - 6, we find (x - 3)(x + 2). So, the denominator equals zero when x = 3 or x = -2.

Q: What does it mean if the denominator equals zero at a particular point?

If the denominator equals zero at x = a, then f(x) will have either a vertical asymptote or a removable discontinuity at x = a, depending on whether g(x) can be factored into (x - a) times another polynomial.

Q: How can we determine the graph of f(x) from the given choices?

We compare the given graph choices to the information we have. If a graph choice has a vertical asymptote or removable discontinuity at x = 3 or x = -2, it could be a possible graph of f(x). Choice C is the only one that aligns with these conditions.

Summary & Key Takeaways

• The function is defined as f(x) = g(x) / (x^2 - x - 6), where g(x) is a polynomial.

• The denominator of f(x), x^2 - x - 6, can be factored into (x - 3)(x + 2).

• The function f(x) will have a vertical asymptote or a removable discontinuity at x = 3 or x = -2, depending on whether g(x) can be factored into (x - 3) times another polynomial.

• Four graph choices are given, and only choice C aligns with the information provided.