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

Introduction

In this comprehensive exploration, we delve into the intricacies of the rational function $f(x) = \frac{x^2 + x - 12}{-x^2 - x + 2}$. Our primary objective is to meticulously determine the domain of this function, pinpoint its vertical asymptotes, and ascertain its horizontal asymptotes. These attributes are fundamental in understanding the behavior and graphical representation of rational functions. By systematically analyzing the numerator and denominator, we can unravel the function's key characteristics and gain a deeper understanding of its mathematical properties.

Determining the Domain

The domain of a function encompasses all possible input values (x-values) for which the function yields a defined output. For rational functions, the domain is restricted by values that make the denominator equal to zero, as division by zero is undefined. Therefore, to find the domain of $f(x)$, we must identify the values of $x$ that satisfy the equation $-x^2 - x + 2 = 0$. This quadratic equation can be solved by factoring, completing the square, or using the quadratic formula.

Factoring the quadratic expression, we have:

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

Setting this expression equal to zero, we get:

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

This equation holds true when either $x + 2 = 0$ or $x - 1 = 0$. Solving these linear equations yields $x = -2$ and $x = 1$. These are the values that make the denominator zero and 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 $(-\infty, -2) \cup (-2, 1) \cup (1, \infty)$. Understanding the domain is crucial as it dictates the possible inputs for our function and influences the presence of vertical asymptotes.

Identifying Vertical Asymptotes

Vertical asymptotes are vertical lines that a function approaches but never touches or crosses. They occur at $x$-values where the denominator of a rational function equals zero, and the numerator does not. We have already determined that the denominator of $f(x)$ is zero when $x = -2$ and $x = 1$. Now, we need to check if the numerator is also zero at these points. If the numerator is non-zero, then we have a vertical asymptote.

The numerator of $f(x)$ is $x^2 + x - 12$. Let's evaluate this at $x = -2$ and $x = 1$:

For $x = -2$:

$(-2)^2 + (-2) - 12 = 4 - 2 - 12 = -10$.

For $x = 1$:

$(1)^2 + (1) - 12 = 1 + 1 - 12 = -10$.

Since the numerator is non-zero at both $x = -2$ and $x = 1$, we can conclude that there are vertical asymptotes at these values. The vertical asymptotes are the lines $x = -2$ and $x = 1$. These lines indicate where the function's value approaches infinity or negative infinity, providing key insights into the function's behavior.

Determining Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as $x$ approaches positive or negative infinity. To find horizontal asymptotes, we compare the degrees of the numerator and denominator polynomials.

In our function, $f(x) = \frac{x^2 + x - 12}{-x^2 - x + 2}$, both the numerator and the denominator are quadratic polynomials (degree 2). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

The leading coefficient of the numerator ($x^2 + x - 12$) is 1. The leading coefficient of the denominator ($-x^2 - x + 2$) is -1.

Therefore, the horizontal asymptote is given by:

$y = \frac{1}{-1} = -1$.

Thus, the horizontal asymptote is the line $y = -1$. This means that as $x$ approaches positive or negative infinity, the function $f(x)$ approaches the value -1. The horizontal asymptote provides valuable information about the long-term behavior of the function.

Comprehensive Analysis and Conclusion

In summary, for the rational function $f(x) = \frac{x^2 + x - 12}{-x^2 - x + 2}$, we have determined the following:

• Domain: The domain is all real numbers except $x = -2$ and $x = 1$, expressed in interval notation as $(-\infty, -2) \cup (-2, 1) \cup (1, \infty)$.
• Vertical Asymptotes: The vertical asymptotes are the lines $x = -2$ and $x = 1$.
• Horizontal Asymptote: The horizontal asymptote is the line $y = -1$.

Understanding these characteristics allows us to sketch the graph of the function and comprehend its behavior. The domain restricts the possible $x$-values, the vertical asymptotes show where the function approaches infinity, and the horizontal asymptote describes the function's behavior as $x$ tends to infinity.

By carefully analyzing the domain, vertical asymptotes, and horizontal asymptotes of the given rational function, we gain a complete understanding of its properties and behavior. This analytical approach is fundamental in the study of functions and their graphical representations.