Slant Asymptote Find The Equation Of F(x) = (x^2 + X + 4) / (x - 1)

Jul 11, 2025 by ADMIN 68 views

Finding the Slant Asymptote of a Rational Function: A Comprehensive Guide

In mathematics, understanding the behavior of functions, especially rational functions, is crucial. One key aspect of this understanding involves identifying asymptotes, which are lines that a curve approaches but never quite touches. Among different types of asymptotes, slant asymptotes, also known as oblique asymptotes, are particularly interesting. This article will provide a comprehensive guide on how to find the equation of the slant asymptote of the function f(x) = (x^2 + x + 4) / (x - 1). We will explore the underlying principles, the step-by-step process, and the rationale behind each step, ensuring a clear and thorough understanding of this concept.

Understanding Slant Asymptotes

Before diving into the specifics of the given function, it's important to grasp the concept of slant asymptotes. A slant asymptote occurs in a rational function when the degree of the numerator is exactly one greater than the degree of the denominator. In simpler terms, if you have a rational function in the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials, a slant asymptote exists if the highest power of x in P(x) is one more than the highest power of x in Q(x). This condition is essential because it indicates that as x approaches positive or negative infinity, the function will behave like a linear function, which is the slant asymptote. Identifying a slant asymptote involves polynomial long division or synthetic division, which helps to rewrite the rational function in a form that reveals the linear asymptote. The quotient obtained from the division represents the equation of the slant asymptote. Understanding this fundamental concept is crucial for solving problems related to rational functions and their asymptotic behavior.

Step-by-Step Guide to Finding the Slant Asymptote

Now, let's apply this knowledge to the given function, f(x) = (x^2 + x + 4) / (x - 1). To find the equation of the slant asymptote, we will follow a step-by-step process:

Step 1: Verify the Condition for a Slant Asymptote

The first step is to ensure that a slant asymptote actually exists. As mentioned earlier, this happens when the degree of the numerator is exactly one greater than the degree of the denominator. In our function, the numerator x^2 + x + 4 has a degree of 2 (the highest power of x is 2), and the denominator x - 1 has a degree of 1 (the highest power of x is 1). Since 2 is one greater than 1, a slant asymptote exists. This verification is crucial because it confirms that proceeding with polynomial division is the correct approach. Without this confirmation, one might waste time on an unnecessary process. Therefore, always check this condition first to streamline your problem-solving approach.

Step 2: Perform Polynomial Long Division

The next step is to perform polynomial long division. This process will divide the numerator (x^2 + x + 4) by the denominator (x - 1). The setup for long division is similar to that of numerical long division. We write the divisor (x - 1) outside the division bracket and the dividend (x^2 + x + 4) inside. The goal is to find a quotient and a remainder. Start by dividing the leading term of the dividend (x^2) by the leading term of the divisor (x), which gives x. Multiply the entire divisor (x - 1) by this quotient term (x) and subtract the result from the dividend. Bring down the next term from the dividend and repeat the process until the degree of the remainder is less than the degree of the divisor. The quotient obtained in this process will be the non-fractional part of the rewritten function, which represents the slant asymptote.

Step 3: Identify the Quotient

After performing the long division, we will obtain a quotient and a remainder. The quotient represents the equation of the slant asymptote. In the division process for f(x) = (x^2 + x + 4) / (x - 1), we find that x^2 + x + 4 divided by x - 1 gives a quotient of x + 2 and a remainder of 6. This means we can rewrite the function as f(x) = (x + 2) + 6/(x - 1). The term x + 2 is the quotient we are interested in, as it represents the linear function that the original function approaches as x goes to infinity or negative infinity. This quotient is the key to defining the slant asymptote.

Step 4: Write the Equation of the Slant Asymptote

Based on the quotient obtained from the long division, we can now write the equation of the slant asymptote. In our case, the quotient is x + 2. Therefore, the equation of the slant asymptote is y = x + 2. This equation represents a straight line that the function f(x) approaches as x gets very large or very small. The remainder term, 6/(x - 1), becomes negligible as x approaches infinity, leaving the linear function y = x + 2 as the dominant behavior of the function. Thus, the slant asymptote provides valuable information about the long-term behavior of the rational function.

Detailed Polynomial Long Division

To further illustrate the process, let's perform the polynomial long division in detail:

Set up the division:

      _____________
x - 1 | x^2 + x + 4

Divide the first term of the dividend (x^2) by the first term of the divisor (x): x^2 / x = x
```
      x __________
x - 1 | x^2 + x + 4
```

Multiply the divisor (x - 1) by x: x(x - 1) = x^2 - x

      x __________
x - 1 | x^2 + x + 4
	  x^2 - x

Subtract the result from the dividend: (x^2 + x + 4) - (x^2 - x) = 2x + 4

      x __________
x - 1 | x^2 + x + 4
	  x^2 - x
	  -------
	   2x + 4

Divide the first term of the new dividend (2x) by the first term of the divisor (x): 2x / x = 2
```
      x + 2 ______
x - 1 | x^2 + x + 4
	  x^2 - x
	  -------
	   2x + 4
```

Multiply the divisor (x - 1) by 2: 2(x - 1) = 2x - 2

      x + 2 ______
x - 1 | x^2 + x + 4
	  x^2 - x
	  -------
	   2x + 4
	   2x - 2

Subtract the result from the current dividend: (2x + 4) - (2x - 2) = 6

      x + 2 ______
x - 1 | x^2 + x + 4
	  x^2 - x
	  -------
	   2x + 4
	   2x - 2
	   ------
	       6

Thus, the quotient is x + 2, and the remainder is 6. This confirms our earlier result and provides a detailed walkthrough of the long division process. This detailed explanation helps to clarify any confusion and reinforces the method for finding the slant asymptote.

Conclusion

In conclusion, the equation of the slant asymptote for the function f(x) = (x^2 + x + 4) / (x - 1) is y = x + 2. This result was obtained by first verifying the existence of a slant asymptote, performing polynomial long division, identifying the quotient, and then writing the equation of the asymptote. Understanding how to find the equation of the slant asymptote is a fundamental skill in the study of rational functions, providing insights into the function's behavior as x approaches infinity. The step-by-step guide presented here should serve as a valuable resource for anyone looking to master this concept. By following these steps, you can confidently find the equation of the slant asymptote for any rational function where the degree of the numerator is one greater than the degree of the denominator. This skill is not only important for academic purposes but also has practical applications in various fields where rational functions are used to model real-world phenomena.

Final Answer

The final answer is (A) y = x + 2.