Graphs of rational functions: vertical asymptotes | High School Math | Khan Academy | Summary and Q&A

TL;DR

Determine the possible graph of a polynomial function based on the given information about its denominator and potential vertical asymptotes or removable discontinuities.

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.

Transcript

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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.