Domain, Vertical Asymptotes, And Horizontal Asymptotes Of F(x) = (x^2 + X - 12) / (-x^2 - X + 2)

Jul 10, 2025 by ADMIN 97 views

In this comprehensive exploration, we delve into the intricacies of the rational function f(x) = (x^2 + x - 12) / (-x^2 - x + 2). Our focus will be on meticulously determining the domain, vertical asymptotes, and horizontal asymptotes of this function. These characteristics are crucial for understanding the behavior and graphical representation of rational functions, which are fundamental concepts in mathematics, particularly in calculus and analysis. Understanding these aspects allows us to sketch the graph of the function accurately and predict its behavior for different input values.

Determining the Domain of f(x)

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the primary concern is to identify values of x that would make the denominator equal to zero, as division by zero is undefined. Therefore, to find the domain of f(x) = (x^2 + x - 12) / (-x^2 - x + 2), we need to determine the values of x for which the denominator, -x^2 - x + 2, is not equal to zero.

First, let's find the roots of the denominator by setting it equal to zero:

-x^2 - x + 2 = 0

To make the equation easier to solve, we can multiply the entire equation by -1:

x^2 + x - 2 = 0

Now, we can factor the quadratic equation:

(x + 2)(x - 1) = 0

This gives us two solutions:

x = -2 and x = 1

These are the values of x that make the denominator zero, and thus, they must be excluded from the domain. Therefore, the domain of f(x) is all real numbers except x = -2 and x = 1. In interval notation, this can be expressed as:

Domain: (-∞, -2) ∪ (-2, 1) ∪ (1, ∞)

Understanding the domain is the first step in analyzing the function. It tells us where the function is valid and where it is undefined, which is critical for identifying vertical asymptotes and understanding the function's overall behavior. The domain sets the stage for further analysis, allowing us to focus on the regions where the function is well-behaved and continuous. Ignoring the domain can lead to incorrect interpretations of the function's behavior and graph.

Identifying Vertical Asymptotes

Vertical asymptotes occur at x-values where the function approaches infinity (or negative infinity). In rational functions, these often occur where the denominator is zero, and the numerator is non-zero. We've already found the values that make the denominator zero: x = -2 and x = 1. Now, we need to check if these values also make the numerator zero.

The numerator is x^2 + x - 12. Let's factor it:

x^2 + x - 12 = (x + 4)(x - 3)

Now, we can see that the numerator is zero when x = -4 or x = 3. Since these values are different from x = -2 and x = 1, we can conclude that the function will approach infinity at x = -2 and x = 1. Therefore, these are our vertical asymptotes.

To confirm this, we can analyze the behavior of the function as x approaches these values. As x approaches -2 from the left, the denominator becomes a small positive number, and the numerator is negative, so f(x) approaches negative infinity. As x approaches -2 from the right, the denominator becomes a small negative number, and the numerator is negative, so f(x) approaches positive infinity. A similar analysis can be done for x = 1.

The vertical asymptotes are crucial for sketching the graph of the function. They act as boundaries that the graph approaches but never crosses. They provide important information about the function's behavior near these points of discontinuity. The presence of vertical asymptotes significantly influences the shape and characteristics of the graph, making them a key aspect of the function's analysis.

Determining Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote of f(x) = (x^2 + x - 12) / (-x^2 - x + 2), we need to examine the limit of the function as x approaches infinity and negative infinity. The general rule for rational functions is to compare the degrees of the numerator and the denominator.

In this case, the degree of the numerator (x^2 + x - 12) is 2, and the degree of the denominator (-x^2 - x + 2) is also 2. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is -1. Therefore, the horizontal asymptote is:

y = 1 / -1 = -1

This means that as x approaches positive or negative infinity, the function f(x) approaches y = -1. To confirm this, we can divide both the numerator and the denominator by the highest power of x, which is x^2:

f(x) = (1 + 1/x - 12/x^2) / (-1 - 1/x + 2/x^2)

As x approaches infinity, the terms 1/x and 1/x^2 approach 0, so the function approaches 1 / -1 = -1.

The horizontal asymptote provides a sense of the long-term behavior of the function. It indicates the value the function tends towards as x becomes very large or very small. The presence of a horizontal asymptote helps in understanding the overall trend of the graph and provides a reference line for the function's values at extreme x values. Identifying the horizontal asymptote is essential for sketching an accurate graph of the function.

Conclusion

In summary, for the function f(x) = (x^2 + x - 12) / (-x^2 - x + 2), we have determined the following:

Domain: (-∞, -2) ∪ (-2, 1) ∪ (1, ∞)
Vertical Asymptotes: x = -2 and x = 1
Horizontal Asymptote: y = -1

These characteristics provide a comprehensive understanding of the function's behavior. The domain tells us where the function is defined, the vertical asymptotes indicate points of discontinuity, and the horizontal asymptote describes the function's behavior as x approaches infinity. By combining this information, we can create an accurate sketch of the function's graph and gain a deeper understanding of its properties. Analyzing the domain, vertical asymptotes, and horizontal asymptotes is a fundamental process in understanding rational functions and their applications in various fields of mathematics and science.

Understanding these core aspects of rational functions allows for a more nuanced analysis of their behavior and graphical representation. The interplay between the domain, vertical asymptotes, and horizontal asymptotes provides a framework for predicting the function's values and trends, making them invaluable tools in mathematical analysis and modeling.