Problem 44 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(39-44,\) (a) find a power function end behavior model for \(f .\) (b) Identify any horizontal asymptotes. $$f(x)=\frac{-x^{4}+2 x^{2}+x-3}{x^{2}-4}$$

Step-by-Step Solution

Answer

The power function end behavior model for \(f\) is \(-x^{2}\). The function \(f\) does not have any horizontal asymptotes.

Step 1: Simplify the function

Rewrite \(f(x)\) as \(-x^{4}+2 x^{2}+x-3\) and \(x^{2}-4\). Because the highest exponent value in the numerator is \(4\) (from \(-x^{4}\)) and in the denominator is \(2\) (from \(x^{2}\)), we can simplify the function to be \(-x^{2} + \frac{2}{x^{2}} + \frac{1}{x^{3}} - \frac{3}{x^{4}}\). We get this form by dividing each term in the numerator by \(x^{2}\).

Step 2: Identify the power function end behavior model

Examining the simplified function, the term with the highest power is \(-x^{2}\). Hence, the power function end behavior model for \(f\) is \(-x^{2}\).

Step 3: Identify the horizontal asymptotes

When \(x\) approaches infinity, the terms \(\frac{2}{x^{2}}\), \(\frac{1}{x^{3}}\), and \(\frac{3}{x^{4}}\) approach \(0\). This simplifies our function to \(-x^{2}\). Therefore, the function \(f\) does not have any horizontal asymptotes as \(-x^{2}\) tends to \(-\infty\) as \(x\) tends to \(\infty\) and \(x^{2}\) tends to \(\infty\) as \(x\) tends to \(-\infty\).

Key Concepts

End Behavior

The end behavior of a function describes what happens to the values of a function as the input grows very large in the positive or negative direction. For rational functions, which involve polynomials in both the numerator and the denominator, end behavior is often determined by the largest degree terms. In the exercise provided, the focus is on

• the term of the highest degree in the numerator: \(-x^4\)
• and the highest degree in the denominator: \(x^2\)

By comparing these expressions, we recognize that as \( x \) becomes very large, terms with smaller powers diminish. Thus, the term with the largest power controls the function's end behavior. Here, the \(-x^2\) is the critical term, indicating that as \(x\) goes to either positive or negative infinity, the end behavior resembles that of the power function \(-x^2\). This means that \(f(x)\) will tend to \(-\infty\) as \(x\) tends to \(+\infty\), which matches our end behavior model.

Horizontal Asymptotes

Horizontal asymptotes are lines that the graph of a function approaches as \(x\) progresses towards \(+\infty\) or \(-\infty\). They show up in rational functions depending on the degrees of the polynomials in the numerator and denominator. For this problem, the definition of horizontal asymptote depends on the comparison between

• the degree of the polynomial in the numerator
• and the degree of the polynomial in the denominator

If the numerator's degree is less than the denominator's, the horizontal asymptote is \(y = 0\). For equal degrees, the horizontal asymptote is the ratio of the leading coefficients.
However, in this problem, the numerator's degree is greater than that of the denominator. This leads to no horizontal asymptote. Instead, the function grows without bound in either the positive or negative direction, mimicking a power polynomial function such as \(-x^2\). Consequently, the function behaves like a curve that never flattens out to align with a horizontal line.

Power Functions

Power functions are simpler polynomials of the form \(ax^n\), where \(a\) is a constant and \(n\) a non-negative integer. These functions are straightforward because their graphs are determined solely by the term \(ax^n\). For such functions, understanding them helps explain complex rational functions' behaviors, especially at the extremes (very large or very small \(x\) values).
In this exercise, examining the simplified form identifies \(-x^2\) as the power function that describes the end behavior. This power function helps us understand how \(f(x)\) behaves as \(x\) becomes very large or very small. Simple power functions like \(-x^2\) show us a predictable pattern:

• As \(x\) approaches \(\pm\infty\), the values of \(-x^2\) tend to \(-\infty\).
• This pattern sheds light on the leading polynomial's dominance in determining the overall end behavior.

Thus, power functions serve as a navigational tool, predicting behaviors of more intricate rational functions.