Identifying Rational Functions With Horizontal Asymptotes At Y=1

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In the realm of mathematics, rational functions play a crucial role. These functions, expressed as the quotient of two polynomials, exhibit fascinating behaviors, particularly concerning their asymptotes. Among these, horizontal asymptotes offer valuable insights into the function's end behavior. In this article, we will delve into the concept of horizontal asymptotes and explore how to identify a rational function with a horizontal asymptote at y = 1. We will dissect the key principles governing horizontal asymptotes, enabling you to confidently determine the correct function from a given set of options.

Delving into Rational Functions

At their core, rational functions are functions that can be written as a ratio of two polynomials. Mathematically, this can be expressed as f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomial functions. The behavior of rational functions is largely dictated by the degrees and leading coefficients of these polynomials. One of the most revealing aspects of a rational function's behavior is its asymptotes. An asymptote is a line that the graph of the function approaches but does not actually touch or cross. There are three main types of asymptotes: vertical, horizontal, and oblique (or slant). In this discussion, our primary focus will be on horizontal asymptotes.

Horizontal Asymptotes: A Key to End Behavior

Horizontal asymptotes provide a glimpse into the end behavior of a rational function. They describe what happens to the function's output (y-value) as the input (x-value) approaches positive or negative infinity. In simpler terms, they tell us where the function is heading as we move far away from the origin along the x-axis. The existence and location of horizontal asymptotes are determined by comparing the degrees of the polynomials in the numerator and the denominator.

To determine the horizontal asymptote of a rational function f(x) = P(x) / Q(x), we consider three scenarios:

  1. Degree of P(x) < Degree of Q(x): In this case, the horizontal asymptote is always y = 0. This is because, as x becomes very large, the denominator grows much faster than the numerator, causing the fraction to approach zero.
  2. Degree of P(x) = Degree of Q(x): Here, the horizontal asymptote is y = a/b, where a is the leading coefficient of P(x) and b is the leading coefficient of Q(x). The leading coefficient is the number that multiplies the highest power of x in the polynomial. As x becomes very large, the terms with the highest powers dominate, and their ratio determines the horizontal asymptote.
  3. Degree of P(x) > Degree of Q(x): In this scenario, there is no horizontal asymptote. Instead, there may be an oblique (or slant) asymptote, which is a diagonal line that the function approaches as x goes to infinity.

Identifying a Rational Function with a Horizontal Asymptote at y = 1

Now, let's apply our understanding of horizontal asymptotes to the specific problem at hand. We are tasked with identifying a rational function h(x) that is continuous and has a horizontal asymptote at y = 1. This means that as x approaches positive or negative infinity, the value of h(x) gets closer and closer to 1.

Based on our earlier discussion, we know that a horizontal asymptote at y = 1 occurs when the degree of the numerator and the degree of the denominator are equal, and the ratio of their leading coefficients is 1. This is the key criterion we will use to evaluate potential functions.

Analyzing the Given Option

Let's consider the option provided:

A. h(x) = (x² - 16) / (x² + 16)

To determine if this function has a horizontal asymptote at y = 1, we need to examine the degrees and leading coefficients of the numerator and denominator.

Step-by-Step Analysis

  1. Identify the degrees: The numerator, x² - 16, has a degree of 2 (the highest power of x is 2). The denominator, x² + 16, also has a degree of 2.
  2. Compare the degrees: Since the degrees of the numerator and denominator are equal, we know that there is a horizontal asymptote.
  3. Determine the leading coefficients: The leading coefficient of the numerator is 1 (the coefficient of x²). The leading coefficient of the denominator is also 1 (the coefficient of x²).
  4. Calculate the ratio: The ratio of the leading coefficients is 1/1 = 1.
  5. Identify the horizontal asymptote: Since the ratio of the leading coefficients is 1, the horizontal asymptote is y = 1.

Conclusion for Option A

Based on our analysis, the function h(x) = (x² - 16) / (x² + 16) has a horizontal asymptote at y = 1. This aligns with the given requirement. Additionally, the function is continuous because the denominator x² + 16 is never zero for any real value of x.

Additional Considerations for Identifying Rational Functions

While analyzing the degrees and leading coefficients is crucial for determining horizontal asymptotes, there are other aspects to consider when working with rational functions:

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of the rational function is zero, and the numerator is non-zero. These values indicate points where the function approaches infinity or negative infinity. To find vertical asymptotes, set the denominator equal to zero and solve for x. It's important to note that if a factor is common to both the numerator and denominator, it may result in a hole (removable discontinuity) rather than a vertical asymptote.

Holes (Removable Discontinuities)

A hole occurs in the graph of a rational function when a factor is common to both the numerator and denominator. This factor can be canceled out, but the original function is undefined at the value of x that makes this factor zero. This results in a hole in the graph at that point. To find holes, factor both the numerator and denominator, cancel any common factors, and then set the canceled factor equal to zero and solve for x. This value of x corresponds to the location of the hole.

Intercepts

Intercepts are the points where the graph of the function crosses the x-axis (x-intercepts) or the y-axis (y-intercept*). To find the x-intercepts, set f(x) = 0 and solve for x. This is equivalent to setting the numerator of the rational function equal to zero and solving for x. To find the y-intercept, set x = 0 and evaluate f(0).

Summarizing Key Concepts

To effectively work with rational functions, it is essential to grasp the following key concepts:

  • Rational Function: A function that can be expressed as the ratio of two polynomials.
  • Horizontal Asymptote: A horizontal line that the graph of the function approaches as x approaches positive or negative infinity. Determined by comparing the degrees and leading coefficients of the numerator and denominator.
  • Vertical Asymptote: A vertical line at x-values where the denominator is zero and the numerator is non-zero.
  • Hole (Removable Discontinuity): A point where the function is undefined due to a common factor in the numerator and denominator.
  • Intercepts: Points where the graph crosses the x-axis (x-intercepts) or the y-axis (y-intercept*).

By mastering these concepts, you will be well-equipped to analyze and understand the behavior of rational functions, including identifying their asymptotes, holes, and intercepts.

Conclusion

In this comprehensive exploration, we have dissected the concept of horizontal asymptotes in rational functions. We established that a rational function h(x) has a horizontal asymptote at y = 1 if the degree of the numerator is equal to the degree of the denominator, and the ratio of their leading coefficients is 1. By applying this knowledge, we successfully analyzed the given option h(x) = (x² - 16) / (x² + 16) and confirmed that it indeed satisfies the condition of having a horizontal asymptote at y = 1. Furthermore, we broadened our understanding by discussing vertical asymptotes, holes, and intercepts, providing a holistic view of rational function behavior. This understanding empowers you to confidently identify and analyze rational functions in various mathematical contexts.

This analytical approach is critical for success in advanced mathematics, where a deep understanding of function behavior is paramount. As you continue your mathematical journey, remember the principles we've discussed here, and you'll find yourself well-prepared to tackle complex problems involving rational functions and their asymptotes.