Analyzing End Behavior Of F(x) = (x^2 - 100) / (x^2 - 3x - 4)

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In mathematics, particularly in the study of functions, understanding end behavior is crucial for grasping the overall characteristics of a function. End behavior describes what happens to the function's output (y-values) as the input (x-values) approaches positive or negative infinity. This is especially important for rational functions, which are functions expressed as the ratio of two polynomials. In this article, we will delve into the end behavior of the rational function f(x) = (x^2 - 100) / (x^2 - 3x - 4), providing a comprehensive analysis and explanation.

Defining End Behavior

Before we dive into the specifics of our function, let's clarify what we mean by "end behavior." Imagine tracing a function's graph with your finger. As your finger moves further and further to the right (approaching positive infinity) or to the left (approaching negative infinity) along the x-axis, what happens to the graph? Does it shoot up towards positive infinity, plummet down towards negative infinity, or level off and approach a specific value? The answer to this question describes the function's end behavior.

For rational functions, end behavior is primarily determined by the degrees of the polynomials in the numerator and denominator. The degree of a polynomial is the highest power of the variable (usually x) in the polynomial. Let's look at a few general rules:

  • If the degree of the numerator is less than the degree of the denominator: The function will approach 0 as x approaches both positive and negative infinity. In other words, the x-axis (y = 0) acts as a horizontal asymptote.
  • If the degree of the numerator is equal to the degree of the denominator: The function will approach a non-zero constant as x approaches both positive and negative infinity. This constant is the ratio of the leading coefficients (the coefficients of the terms with the highest power of x) of the numerator and denominator. The horizontal asymptote is y = (ratio of leading coefficients).
  • If the degree of the numerator is greater than the degree of the denominator: The function will approach either positive or negative infinity as x approaches positive or negative infinity. There will be no horizontal asymptote, but there may be a slant (oblique) asymptote.

Analyzing the Function f(x) = (x^2 - 100) / (x^2 - 3x - 4)

Now, let's apply these concepts to our specific function:

f(x) = (x^2 - 100) / (x^2 - 3x - 4)

Step 1: Identify the Degrees of the Polynomials

The numerator, x^2 - 100, is a polynomial of degree 2 (the highest power of x is 2). The denominator, x^2 - 3x - 4, is also a polynomial of degree 2.

Step 2: Compare the Degrees

Since the degrees of the numerator and denominator are equal (both are 2), we know that the function will approach a non-zero constant as x approaches positive and negative infinity. This means the function has a horizontal asymptote.

Step 3: Determine the Ratio of Leading Coefficients

The leading coefficient of the numerator (x^2 - 100) is 1 (the coefficient of the x^2 term). The leading coefficient of the denominator (x^2 - 3x - 4) is also 1 (the coefficient of the x^2 term). The ratio of the leading coefficients is 1/1 = 1.

Step 4: State the End Behavior

Therefore, as x approaches positive infinity (x → ∞), f(x) approaches 1. Similarly, as x approaches negative infinity (x → -∞), f(x) also approaches 1. This indicates that the horizontal asymptote of the function is y = 1.

Graphical Representation

Visualizing the graph of f(x) = (x^2 - 100) / (x^2 - 3x - 4) can further solidify our understanding of its end behavior. If you were to plot this function, you would observe that as you move further to the left and right along the x-axis, the graph gets closer and closer to the horizontal line y = 1, without ever actually touching it. This graphical representation confirms our analytical determination of the end behavior.

Additional Insights: Vertical Asymptotes and Holes

While the end behavior focuses on what happens as x approaches infinity, it's also helpful to consider other aspects of the function's behavior. Specifically, we can explore vertical asymptotes and holes.

Vertical Asymptotes occur where the denominator of the rational function equals zero. To find them, we set the denominator equal to zero and solve for x:

  • x^2 - 3x - 4 = 0
  • (x - 4)(x + 1) = 0
  • x = 4 or x = -1

This means there are vertical asymptotes at x = 4 and x = -1. The function will approach positive or negative infinity as x gets closer to these values.

Holes occur when a factor cancels out from both the numerator and denominator. In our case, we can factor the numerator as a difference of squares:

  • f(x) = (x^2 - 100) / (x^2 - 3x - 4) = ((x - 10)(x + 10)) / ((x - 4)(x + 1))

There are no common factors to cancel, so there are no holes in this function.

In Summary, the end behavior of the rational function f(x) = (x^2 - 100) / (x^2 - 3x - 4) is that the function approaches 1 as x approaches both positive and negative infinity. This is because the degrees of the numerator and denominator are equal, and the ratio of their leading coefficients is 1. Additionally, the function has vertical asymptotes at x = 4 and x = -1, and no holes. Understanding End Behavior of Rational Functions is important for calculus and precalculus. Analyzing Rational Functions include finding asymptotes.

Now, let's address the original question directly: Which statement accurately describes the end behavior of the function f(x) = (x^2 - 100) / (x^2 - 3x - 4)?

We've already determined that the function approaches 1 as x approaches both positive and negative infinity. Let's evaluate the given options in light of this understanding.

The Question:

Which statement describes the end behavior of the function?

f(x) = (x^2 - 100) / (x^2 - 3x - 4)

The Options:

  • A. The function approaches 0 as x approaches -∞ and ∞.
  • B. [The actual statement for option B was not provided, but we will analyze based on our understanding]

Analysis:

  • Option A: This statement is incorrect. We've shown through our analysis that the function approaches 1, not 0, as x approaches -∞ and ∞. Understanding end behavior is crucial to answering this question.
  • Option B: Based on our previous analysis, we know the correct statement should indicate that the function approaches 1 as x approaches both positive and negative infinity. If option B stated something along these lines, it would be the correct answer. For instance, if Option B stated: "The function approaches 1 as x approaches -∞ and ∞," then Option B would be the correct answer.

Conclusion:

The correct statement describing the end behavior of the function f(x) = (x^2 - 100) / (x^2 - 3x - 4) is that it approaches 1 as x approaches both negative and positive infinity. Therefore, option A is incorrect, and option B (if it stated the function approaches 1) would be the correct choice. This detailed analysis of rational functions provides a clear understanding of their behavior. Identifying End Behavior requires careful examination of the degrees and coefficients.

Understanding end behavior is not just an academic exercise; it has practical applications in various fields. In calculus, end behavior helps in determining the convergence or divergence of integrals. In real-world modeling, it can provide insights into the long-term trends of a system. For example, in population growth models, end behavior can tell us whether a population will stabilize, grow indefinitely, or decline to extinction. Understanding end behavior of functions helps to make predictions.

Importance in Graphing

End behavior plays a pivotal role in sketching the graph of a function. By knowing how a function behaves as x approaches infinity, we can accurately depict the overall shape of the graph. This is especially true for rational functions, where end behavior, along with vertical asymptotes and intercepts, provides a framework for constructing the graph. Graphing functions requires knowledge of end behavior.

Applications in Calculus

In calculus, the concept of limits is fundamental, and end behavior is essentially a limit as x approaches infinity. This understanding is crucial for determining the existence of horizontal asymptotes, which are essential for analyzing the function's global behavior. Calculus and limits are connected to end behavior.

Practical Examples

Consider a rational function that models the concentration of a drug in the bloodstream over time. The end behavior of this function can tell us the long-term concentration of the drug, which is vital information for determining appropriate dosages. Similarly, in economics, rational functions can model cost-benefit ratios, and the end behavior can indicate the long-term efficiency of a particular strategy. Examples of rational functions are common in real-world applications.

Techniques for Determining End Behavior

Beyond the method of comparing degrees, there are other techniques for determining end behavior. For instance, we can use long division to rewrite the rational function as the sum of a polynomial and a proper rational function (where the degree of the numerator is less than the degree of the denominator). The end behavior of the original function is then the same as the end behavior of the polynomial. Different techniques can be used to find end behavior.

Another technique involves dividing both the numerator and denominator by the highest power of x present in the function. This simplifies the expression and makes it easier to see what happens as x becomes very large. Simplifying expressions can help determine end behavior.

In conclusion, understanding end behavior is crucial for analyzing the characteristics of functions, particularly rational functions. It allows us to predict the function's long-term trends, sketch its graph accurately, and apply it to various real-world scenarios. The function f(x) = (x^2 - 100) / (x^2 - 3x - 4) serves as an excellent example to illustrate these concepts, demonstrating that as x approaches infinity, the function approaches 1.

The concept of end behavior might seem abstract at first, but it is a powerful tool for understanding the overall nature of functions. By examining end behavior, we gain insights into the function's long-term trends, its potential for growth or decay, and its relationship to other mathematical concepts. The analysis we've conducted on f(x) = (x^2 - 100) / (x^2 - 3x - 4) highlights the importance of end behavior in understanding the characteristics and behavior of rational functions. Final thoughts on end behavior emphasize its importance.

By understanding end behavior, we can predict the long-term behavior of the function and make informed decisions based on its properties. Long-term behavior can be predicted with end behavior analysis.