(x^3+x^2+x+2)/(x^2-1)

5 min read Jun 17, 2024

Analyzing the Rational Function: (x³ + x² + x + 2) / (x² - 1)

This article will explore the rational function (x³ + x² + x + 2) / (x² - 1). We will analyze its key features, including:

Domain and Range: Identifying the values for which the function is defined and the possible output values.
Vertical Asymptotes: Understanding where the function approaches infinity.
Horizontal Asymptotes: Examining the behavior of the function as x approaches positive and negative infinity.
Holes: Determining if there are any points where the function is undefined but can be "filled in".
Intercepts: Finding where the graph of the function crosses the x-axis and y-axis.

Domain and Range

The function is defined for all values of x except where the denominator is zero.

x² - 1 = 0 implies x = 1 or x = -1. Therefore, the domain of the function is (-∞, -1) U (-1, 1) U (1, ∞).

The range of the function is more complex to determine. We can analyze the behavior of the function near the vertical asymptotes and as x approaches positive and negative infinity to get an idea of the range.

Vertical Asymptotes

The function has vertical asymptotes at x = 1 and x = -1, since the denominator becomes zero at these points.

Horizontal Asymptotes

To find the horizontal asymptotes, we compare the degrees of the numerator and denominator. The degree of the numerator (3) is greater than the degree of the denominator (2). This indicates that the function has no horizontal asymptotes. Instead, it has an oblique asymptote.

To find the equation of the oblique asymptote, we perform long division of the numerator by the denominator:

        x + 1
x² - 1 | x³ + x² + x + 2 
        -(x³ - x)
        ----------
            x² + 2x + 2
            -(x² - 1)
            ---------
               2x + 3

The result of the long division is x + 1 + (2x + 3)/(x² - 1). Therefore, the oblique asymptote is y = x + 1.

Holes

We need to check for potential holes in the graph. This occurs when both the numerator and denominator have a common factor.

We can factor the denominator as (x - 1)(x + 1). The numerator, however, does not have any factors that cancel out the denominator. Hence, there are no holes in the graph.

Intercepts

x-intercepts: To find the x-intercepts, we set the numerator equal to zero: x³ + x² + x + 2 = 0

This equation is difficult to solve analytically. We can use numerical methods or graphing tools to approximate the x-intercepts.
y-intercept: To find the y-intercept, we set x equal to zero: (0³ + 0² + 0 + 2) / (0² - 1) = -2

The y-intercept is (0, -2).

Conclusion

We have analyzed the key features of the rational function (x³ + x² + x + 2) / (x² - 1). We found that it has vertical asymptotes at x = 1 and x = -1, no horizontal asymptotes but an oblique asymptote at y = x + 1, and a y-intercept at (0, -2). We also determined that it has no holes in its graph. Further analysis of the x-intercepts would require numerical methods or graphing tools. By understanding these characteristics, we can accurately sketch the graph of this function.