At Each Of The Following Values Of X X X , Determine Whether G ( X ) = X 2 + 5 X − 14 X 2 + 4 X − 21 G(x) = \frac{x^2 + 5x - 14}{x^2 + 4x - 21} G ( X ) = X 2 + 4 X − 21 X 2 + 5 X − 14 Has A Zero, A Vertical Asymptote, Or A Removable Discontinuity.

Mar 10, 2025 by ADMIN 250 views

**Analyzing Discontinuities in Rational Functions**

Introduction

Rational functions are a fundamental concept in algebra, and understanding their behavior is crucial for solving various mathematical problems. In this article, we will delve into the analysis of discontinuities in rational functions, specifically focusing on the function $g(x) = \frac{x^2 + 5x - 14}{x^2 + 4x - 21}$ . We will examine the behavior of this function at specific values of $x$ to determine whether it has a zero, a vertical asymptote, or a removable discontinuity.

Understanding Discontinuities

Discontinuities in rational functions occur when the denominator is equal to zero, resulting in an undefined value. There are three types of discontinuities:

Zero: A zero occurs when the numerator is equal to zero, but the denominator is not.
Vertical Asymptote: A vertical asymptote occurs when the denominator is equal to zero, but the numerator is not.
Removable Discontinuity: A removable discontinuity occurs when both the numerator and denominator are equal to zero, but the function can be simplified to remove the discontinuity.

Analyzing the Function

To analyze the function $g(x) = \frac{x^2 + 5x - 14}{x^2 + 4x - 21}$ , we need to examine the behavior of the numerator and denominator at specific values of $x$ .

Factoring the Numerator and Denominator

The numerator and denominator can be factored as follows:

$x^2 + 5x - 14 = (x + 7)(x - 2)$

$x^2 + 4x - 21 = (x + 7)(x - 3)$

Finding Zeros

To find the zeros of the function, we need to set the numerator equal to zero and solve for $x$ .

$(x + 7)(x - 2) = 0$

This gives us two possible values for $x$ : $x = -7$ and $x = 2$ .

Finding Vertical Asymptotes

To find the vertical asymptotes, we need to set the denominator equal to zero and solve for $x$ .

$(x + 7)(x - 3) = 0$

This gives us two possible values for $x$ : $x = -7$ and $x = 3$ .

Finding Removable Discontinuities

To find the removable discontinuities, we need to find the values of $x$ where both the numerator and denominator are equal to zero.

$(x + 7)(x - 2) = 0$ and $(x + 7)(x - 3) = 0$

This gives us one possible value for $x$ : $x = -7$ .

Conclusion

In conclusion, the function $g(x) = \frac{x^2 + 5x - 14}{x^2 + 4x - 21}$ has a zero at $x = 2$ , a vertical asymptote at $x = 3$ , and a removable discontinuity at $x = -7$ . Understanding the behavior of rational functions is crucial for solving various mathematical problems, and this analysis provides a comprehensive overview of the discontinuities in this specific function.

Recommendations for Further Study

For further study, we recommend exploring the following topics:

Rational Function Simplification: Learn how to simplify rational functions to remove discontinuities.
Graphing Rational Functions: Understand how to graph rational functions to visualize their behavior.
Analyzing Discontinuities in Other Functions: Explore the behavior of other types of functions, such as polynomial and trigonometric functions.

By following these recommendations, you will gain a deeper understanding of rational functions and their behavior, which will enable you to tackle more complex mathematical problems with confidence.

Glossary of Terms

Rational Function: A function that can be expressed as the ratio of two polynomials.
Discontinuity: A point where a function is not defined.
Zero: A point where the numerator is equal to zero, but the denominator is not.
Vertical Asymptote: A point where the denominator is equal to zero, but the numerator is not.
Removable Discontinuity: A point where both the numerator and denominator are equal to zero, but the function can be simplified to remove the discontinuity.

Introduction

In our previous article, we explored the analysis of discontinuities in rational functions, specifically focusing on the function $g(x) = \frac{x^2 + 5x - 14}{x^2 + 4x - 21}$ . We examined the behavior of this function at specific values of $x$ to determine whether it has a zero, a vertical asymptote, or a removable discontinuity. In this article, we will answer some frequently asked questions related to analyzing discontinuities in rational functions.

Q: What is a rational function?

A rational function is a function that can be expressed as the ratio of two polynomials. It is a fundamental concept in algebra and is used to model various real-world phenomena.

A: What are the different types of discontinuities in rational functions?

There are three types of discontinuities in rational functions:

Zero: A zero occurs when the numerator is equal to zero, but the denominator is not.
Vertical Asymptote: A vertical asymptote occurs when the denominator is equal to zero, but the numerator is not.
Removable Discontinuity: A removable discontinuity occurs when both the numerator and denominator are equal to zero, but the function can be simplified to remove the discontinuity.

Q: How do I find the zeros of a rational function?

To find the zeros of a rational function, you need to set the numerator equal to zero and solve for $x$ . This will give you the values of $x$ where the function has a zero.

A: How do I find the vertical asymptotes of a rational function?

To find the vertical asymptotes of a rational function, you need to set the denominator equal to zero and solve for $x$ . This will give you the values of $x$ where the function has a vertical asymptote.

Q: How do I find the removable discontinuities of a rational function?

To find the removable discontinuities of a rational function, you need to find the values of $x$ where both the numerator and denominator are equal to zero. This will give you the values of $x$ where the function has a removable discontinuity.

Q: What is the difference between a zero and a vertical asymptote?

A zero occurs when the numerator is equal to zero, but the denominator is not. A vertical asymptote occurs when the denominator is equal to zero, but the numerator is not.

A: Can a rational function have both a zero and a vertical asymptote at the same value of $x$ ?

No, a rational function cannot have both a zero and a vertical asymptote at the same value of $x$ . If the numerator and denominator are equal to zero at the same value of $x$ , then the function has a removable discontinuity.

Q: How do I simplify a rational function to remove a removable discontinuity?

To simplify a rational function to remove a removable discontinuity, you need to factor the numerator and denominator and cancel out any common factors.

A: What are some common mistakes to avoid when analyzing discontinuities in rational functions?

Some common mistakes to avoid when analyzing discontinuities in rational functions include:

Not factoring the numerator and denominator correctly
Not solving for the correct values of $x$
Not distinguishing between zeros, vertical asymptotes, and removable discontinuities

By avoiding these common mistakes, you will be able to accurately analyze the discontinuities in rational functions and gain a deeper understanding of this important mathematical concept.

Conclusion

In conclusion, analyzing discontinuities in rational functions is a crucial aspect of algebra that requires a deep understanding of the underlying mathematical concepts. By mastering the techniques and strategies outlined in this article, you will be able to accurately analyze the discontinuities in rational functions and gain a deeper understanding of this important mathematical concept.