Identify The Vertical Asymptote(s) For The Function:$f(x)=\frac{x-3}{x^2+x-12}$A. $x=4$B. $x=-4$C. $x=3$D. $x=3, X=-4$

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Understanding Vertical Asymptotes

In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, causing the function to become undefined at that point. In this article, we will focus on identifying the vertical asymptote(s) for the given function f(x)=x−3x2+x−12f(x)=\frac{x-3}{x^2+x-12}.

The Function and Its Factors

The given function is a rational function, which means it is the ratio of two polynomials. The numerator is a linear polynomial, x−3x-3, while the denominator is a quadratic polynomial, x2+x−12x^2+x-12. To identify the vertical asymptote(s), we need to factor the denominator and find its roots.

f(x) = \frac{x-3}{(x+4)(x-3)}

Factoring the Denominator

The denominator can be factored as (x+4)(x−3)(x+4)(x-3). This means that the function has two possible points where it may have a vertical asymptote: x=−4x=-4 and x=3x=3.

Identifying the Vertical Asymptote(s)

To determine which of these points is a vertical asymptote, we need to check if the numerator is equal to zero at that point. If the numerator is not equal to zero, then the point is a vertical asymptote.

f(-4) = \frac{-4-3}{(-4+4)(-4-3)} = \frac{-7}{0} = \infty

Since the numerator is not equal to zero at x=−4x=-4, this point is a vertical asymptote.

f(3) = \frac{3-3}{(3+4)(3-3)} = \frac{0}{7} = 0

However, since the numerator is equal to zero at x=3x=3, this point is not a vertical asymptote.

Conclusion

In conclusion, the vertical asymptote(s) for the function f(x)=x−3x2+x−12f(x)=\frac{x-3}{x^2+x-12} is x=−4x=-4. This is because the denominator is equal to zero at x=−4x=-4, causing the function to become undefined at that point.

Additional Considerations

It's worth noting that if the numerator and denominator of a rational function have a common factor, then that factor will cancel out, and the function will have a hole at that point instead of a vertical asymptote. However, in this case, the numerator and denominator do not have a common factor, so the function has a vertical asymptote at x=−4x=-4.

Final Answer

Understanding Vertical Asymptotes

In our previous article, we discussed how to identify vertical asymptotes in rational functions. A vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, causing the function to become undefined at that point.

Frequently Asked Questions

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, causing the function to become undefined at that point.

Q: How do I identify vertical asymptotes in a rational function?

A: To identify vertical asymptotes in a rational function, you need to factor the denominator and find its roots. Then, check if the numerator is equal to zero at those points. If the numerator is not equal to zero, then the point is a vertical asymptote.

Q: What if the numerator and denominator have a common factor?

A: If the numerator and denominator have a common factor, then that factor will cancel out, and the function will have a hole at that point instead of a vertical asymptote.

Q: Can a rational function have more than one vertical asymptote?

A: Yes, a rational function can have more than one vertical asymptote. This occurs when the denominator has multiple roots.

Q: How do I determine which points are vertical asymptotes and which are holes?

A: To determine which points are vertical asymptotes and which are holes, you need to check if the numerator is equal to zero at those points. If the numerator is not equal to zero, then the point is a vertical asymptote. If the numerator is equal to zero, then the point is a hole.

Q: Can a rational function have no vertical asymptotes?

A: Yes, a rational function can have no vertical asymptotes. This occurs when the denominator has no roots.

Q: How do I graph a rational function with vertical asymptotes?

A: To graph a rational function with vertical asymptotes, you need to plot the vertical asymptotes on the graph and then plot the function's behavior on either side of the asymptotes.

Example Questions

Q: Identify the vertical asymptote(s) for the function f(x)=x+2x2−4f(x)=\frac{x+2}{x^2-4}.

A: To identify the vertical asymptote(s), we need to factor the denominator and find its roots. The denominator can be factored as (x−2)(x+2)(x-2)(x+2). Then, we check if the numerator is equal to zero at those points. Since the numerator is not equal to zero at x=2x=2 and x=−2x=-2, these points are vertical asymptotes.

Q: Identify the vertical asymptote(s) for the function f(x)=x−1x2+2x+1f(x)=\frac{x-1}{x^2+2x+1}.

A: To identify the vertical asymptote(s), we need to factor the denominator and find its roots. The denominator can be factored as (x+1)2(x+1)^2. Then, we check if the numerator is equal to zero at those points. Since the numerator is not equal to zero at x=−1x=-1, this point is a vertical asymptote.

Conclusion

In conclusion, identifying vertical asymptotes in rational functions is an important concept in mathematics. By following the steps outlined in this article, you can easily identify vertical asymptotes and understand the behavior of rational functions.