Finding Zeros, Vertical Asymptotes, And Holes For The Function F(x)=(x^2-x-6)/(x^2+x-2)

Jul 8, 2025 by Jeany 88 views

Finding Zeros, Vertical Asymptotes, and Holes of Rational Functions

In the realm of mathematics, rational functions, which are essentially fractions with polynomials in the numerator and denominator, present intriguing challenges. Unraveling the behavior of these functions requires a keen eye for detail, particularly when it comes to identifying key features such as zeros, vertical asymptotes, and holes. These characteristics provide valuable insights into the function's graph and its overall behavior. Let's embark on a journey to explore the intricacies of finding these elements, using the function $f(x) = \frac{x^2 - x - 6}{x^2 + x - 2}$ as our guide.

Understanding Zeros, Vertical Asymptotes, and Holes

Before diving into the calculations, it's crucial to grasp the fundamental concepts behind zeros, vertical asymptotes, and holes. These features dictate how a function interacts with the x and y axes and how it behaves at specific points.

Zeros: Zeros, also known as roots or x-intercepts, are the points where the function's value equals zero. Graphically, these are the points where the function's graph intersects the x-axis. To find the zeros of a rational function, we focus on the numerator, as the function becomes zero only when the numerator is zero.
Vertical Asymptotes: Vertical asymptotes are vertical lines that the function approaches but never touches. They occur at x-values where the denominator of the rational function equals zero, but the numerator does not. These asymptotes indicate points where the function's value grows infinitely large (positive or negative) as x approaches the asymptote.
Holes: Holes, or removable discontinuities, are points where both the numerator and denominator of the rational function equal zero. These points represent "gaps" in the function's graph, as the function is undefined at these specific x-values. Holes arise when a factor is common to both the numerator and denominator, which can be canceled out.

Step-by-Step Guide to Finding Zeros, Vertical Asymptotes, and Holes for $f(x) = \frac{x^2 - x - 6}{x^2 + x - 2}$

Now, let's apply these concepts to our given function, $f(x) = \frac{x^2 - x - 6}{x^2 + x - 2}$ , and systematically find its zeros, vertical asymptotes, and holes.

1. Factor the Numerator and Denominator

The cornerstone of finding zeros, vertical asymptotes, and holes lies in factoring the numerator and denominator of the rational function. Factoring allows us to identify common factors, which are crucial for pinpointing holes, and to determine the values that make the numerator or denominator zero. In our case, we have:

Numerator: $x^2 - x - 6$
Denominator: $x^2 + x - 2$

Let's factor these expressions:

Factoring the Numerator: We seek two numbers that multiply to -6 and add up to -1. These numbers are -3 and 2. Therefore, we can factor the numerator as follows:

$x^2 - x - 6 = (x - 3)(x + 2)$
Factoring the Denominator: Similarly, we need two numbers that multiply to -2 and add up to 1. These numbers are 2 and -1. Thus, the denominator factors as:

$x^2 + x - 2 = (x + 2)(x - 1)$

Now, we can rewrite the function in its factored form:

$f(x) = \frac{(x - 3)(x + 2)}{(x + 2)(x - 1)}$

2. Identify Holes

Holes occur when a factor is present in both the numerator and the denominator. These common factors can be canceled out, leading to a simplified form of the function. In our factored form, we observe that the factor $(x + 2)$ appears in both the numerator and the denominator. This indicates the presence of a hole. Specifically, the hole occurs at the x-value that makes this factor equal to zero:

$x + 2 = 0 \Rightarrow x = -2$

To find the y-coordinate of the hole, we substitute this x-value into the simplified form of the function, which is obtained after canceling the common factor:

Simplified function: $f(x) = \frac{x - 3}{x - 1}$ , for $x ≠ -2$

Substituting $x = -2$ into the simplified function:

$f(-2) = \frac{-2 - 3}{-2 - 1} = \frac{-5}{-3} = \frac{5}{3}$

Therefore, there is a hole at the point $(-2, \frac{5}{3})$ .

3. Determine Vertical Asymptotes

Vertical asymptotes arise when the denominator of the rational function equals zero, but the numerator does not. After canceling common factors to identify holes, we focus on the remaining factors in the denominator. In our simplified function, the denominator is $(x - 1)$ . Setting this equal to zero, we find:

$x - 1 = 0 \Rightarrow x = 1$

Since the numerator is not zero at $x = 1$ , we have a vertical asymptote at $x = 1$ .

4. Find Zeros

Zeros, or x-intercepts, occur when the numerator of the rational function equals zero. Again, we consider the simplified form of the function after canceling common factors. In our case, the numerator is $(x - 3)$ . Setting this equal to zero, we get:

$x - 3 = 0 \Rightarrow x = 3$

Thus, the function has a zero at $x = 3$ .

Summarizing the Results

After our step-by-step analysis, we can confidently state the key features of the function $f(x) = \frac{x^2 - x - 6}{x^2 + x - 2}$ :

Zeros: $x = 3$
Vertical Asymptotes: $x = 1$
Holes: $(-2, \frac{5}{3})$

These findings provide a comprehensive understanding of the function's behavior and its graphical representation. The zero indicates where the function crosses the x-axis, the vertical asymptote reveals a point where the function approaches infinity, and the hole signifies a removable discontinuity in the graph.

Conclusion

Finding the zeros, vertical asymptotes, and holes of rational functions is a fundamental skill in mathematics. By systematically factoring the numerator and denominator, identifying common factors, and analyzing the resulting expressions, we can gain valuable insights into the function's behavior and its graphical representation. Understanding these features allows us to sketch the graph of the function accurately and to solve a variety of related problems. Mastering these techniques is essential for success in calculus and other advanced mathematical disciplines. This process not only enhances our understanding of rational functions but also sharpens our problem-solving abilities in a broader mathematical context. Remember, practice is key to mastering these concepts, so work through various examples to solidify your understanding.

Keywords: Zeros, Vertical Asymptotes, Holes, Rational Functions, Factoring, Numerator, Denominator, Simplified Function, Removable Discontinuity, x-intercepts, Graphing.