Problem 76 - Precalculus Mathematics for Calculus

Question

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, \quad g(x)=1-x^{2}$$

Step-by-Step Solution

Answer

Vertical asymptote at x = 1; Both f(x) & g(x) have the same end behavior as -x².

Step 1: Simplify the Rational Function

First, we analyze the rational function \( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \). The numerator is a polynomial of degree 4, and the denominator is a polynomial of degree 2. Simplify by factoring out common factors if possible, but in this expression, there is no obvious simplification by factoring that affects asymptote calculation.

Step 2: Identify the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero but the numerator is not zero. Set the denominator equal to zero: \[(x-1)^2 = 0\]Thus, \(x = 1\). Evaluate the numerator at \(x = 1\):\[-(1)^4 + 2(1)^3 - 2(1) = -1 + 2 - 2 = -1\]Since the numerator is not zero when the denominator is, there is a vertical asymptote at \(x = 1\).

Step 3: Determine the End Behavior of f(x)

The end behavior of \( f(x) \) as \( x \to \pm \infty \) can be found by comparing the degrees of the numerator and denominator. The numerator degree (4) is greater than the denominator degree (2), so the end behavior will resemble the polynomial from the numerator. Thus, as \( x \to \pm \infty \), \( f(x) \to -x^2 \) after polynomial division. The end behavior thus mimics \( g(x) = 1 - x^2 \) as \( -x^2 \).

Step 4: Graph Both Functions to Compare End Behavior

Now graph \( f(x) \) and \( g(x) = 1 - x^2 \) in a large viewing rectangle covering a sufficient range of \( x \). You will observe that despite the different terms, for large \( |x| \), both functions approach similar values demonstrating same end behavior, highlighted especially at extremities of the graph.Finally, note the vertical asymptote of \( f(x) \) at \( x = 1 \).

Key Concepts

Vertical Asymptotes

In rational functions like \( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \), vertical asymptotes occur at the values of \( x \) where the denominator is zero but the numerator is not. Consider the denominator \((x-1)^2\). To find where it is zero, solve:

• \((x-1)^2 = 0\)
• This simplifies to \(x = 1\)

Next, check the numerator at \(x = 1\):

• \(-1 + 2 - 2 = -1\)
• The numerator does not equal zero.

Thus, there is a vertical asymptote at \(x = 1\). When graphing, the function will approach but not touch this line, shooting off to infinity or negative infinity at this point. Understanding vertical asymptotes can help predict and explain the behavior of rational functions near specific values of \( x \).

End Behavior

The end behavior of a rational function describes what happens to the function's value as \( x \) approaches infinity or negative infinity. For \( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \), the degrees of the polynomials in the numerator and the denominator are crucial.

• The numerator has degree 4.
• The denominator has degree 2.

When the degree of the numerator is larger than that of the denominator, the end behavior of the rational function resembles that of the highest-degree term of the numerator. Thus, \( f(x) \) behaves similarly to \(-x^2\) as \( x \to \pm \infty \). This outcome means that the function will mimic the parabolic shape of \(-x^2\), opening downwards and extending towards infinity on both sides of the graph.This aspect of end behavior reveals that even when a rational function seems complex, its long-term behavior can often be visualized as a simpler polynomial function. For \( g(x) = 1 - x^2 \), the function similarly trends downwards with a parabolic arc, confirming their shared end behavior.

Polynomial Division

To fully understand the relationship between \( f(x) \) and \( g(x) \), perform polynomial division on the rational function's expression. Polynomial division helps simplify complex functions and reveal simpler forms that describe their behavior.For \( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \), consider polynomial division as breaking down the complex interaction between numerator and denominator into a more digestible form. After dividing, you'll find the leading term contributes significantly to \( f(x)\)'s end behavior, resulting in \(-x^2\).

• This simplification provides clarity, letting you recognize why \( f(x) \) and \( g(x) = 1-x^2 \) appear similar for large magnitudes of \( x \).
• If the degrees of the polynomial weren't amenable to simplification through division, other analytic methods might be employed such as looking at dominant terms.

Understanding polynomial division not only clarifies the end behavior but also unveils the inner workings of rational functions, providing deeper insights into their graphical representations.