Problem 68 - Algebra and Trigonometry

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

The vertical asymptote is at \(x=1\); both functions approach negative infinity as \(x\) approaches infinity, indicating similar end behavior.

Step 1: Identify Vertical Asymptotes

Rational functions can have vertical asymptotes where the denominator is zero but the numerator is not. In this function, the denominator is \((x-1)^2\). Set it to zero to find potential vertical asymptotes: \(x-1=0\) leading to \(x=1\). Thus, the function has a vertical asymptote at \(x=1\).

Step 2: Analyze the End Behavior of f(x) and g(x)

To determine the end behavior of \(f(x)\), consider the leading term of the numerator \(-x^4\) and denominator \((x-1)^2\). As \(x\) approaches infinity, the leading term \(-x^4\) dominates, suggesting the function behaves similarly to \(-x^4\). This implies \(f(x)\) will approach negative infinity as \(x\) goes to positive or negative infinity. For \(g(x) = 1 - x^2\), as \(x\) approaches infinity, \(g(x)\) approaches negative infinity as well. Both functions have similar end behavior as \(x\) approaches both positive and negative infinity.

Step 3: Graph the Functions f(x) and g(x)

Use graphing software or a graphing calculator to sketch both \(f(x)\) and \(g(x)\). Ensure that the display window includes enough of the \(x\)-axis to cover the asymptotic behavior, particularly near the vertical asymptote at \(x = 1\). Note how \(f(x)\) behaves significantly around \(x = 1\) and how both functions approach similar values as \(x\) moves away from this region towards positive and negative infinity.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are crucial when studying rational functions as they help identify the behavior of a function near certain values of \(x\). To find vertical asymptotes, we look for values of \(x\) that make the denominator zero while keeping the numerator non-zero. For the given function \(f(x) = \frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}\), the denominator is \((x-1)^2\). Setting the denominator equal to zero, we solve \(x-1=0\), which gives \(x=1\). Therefore, \(f(x)\) has a vertical asymptote at \(x=1\).

• Vertical asymptotes occur when the denominator is zero but not the numerator.
• For \(f(x)\), the vertical asymptote is at \(x=1\), meaning the graph will approach an infinitely high or low value near this point.

It is essential to recognize vertical asymptotes as they affect how we sketch the graph of the function.

End Behavior

Understanding end behavior provides insights into how a function behaves as \(x\) becomes very large or very small. For rational functions, the end behavior is often determined by the leading terms in the numerator and denominator.
For \(f(x)=\frac{-x^{4}+2x^{3}-2x}{(x-1)^{2}}\), the leading term in the numerator is \(-x^4\), and in the denominator, it is \((x-1)^2\). As \(x\) approaches infinity, \(f(x)\) behaves similarly to \(-x^4\). Both \(f(x)\) and \(g(x)=1-x^{2}\) tend towards negative infinity as \(x\) approaches both positive and negative infinity.

• The leading term in \(f(x)\) dictates that it approaches negative infinity for very large positive or negative \(x\).
• Similarly, \(g(x)\) approaches negative infinity, indicating similar end behavior for both functions.

Understanding end behavior is important when predicting how a graph will look beyond a localized area.

Graphing Functions

Graphing rational functions like \(f(x)\) and \(g(x)\) gives a visual representation of the function's behavior. When graphing, it's important to consider not only the function's general shape but also key features such as intercepts, asymptotes, and end behavior.
Using graphing software or a calculator, plot \(f(x)\) and \(g(x)\) over a large enough viewing range to capture significant features. Pay special attention around \(x=1\), noting that \(f(x)\) shows drastic changes due to the vertical asymptote. As \(x\) moves towards positive and negative infinity, observe that both functions display similar end behavior, confirming the predictions.

• Ensure your graphing tool is set to cover a sufficient range to see the behavior near asymptotes and at endpoints.
• Notice that even with different expressions, \(f(x)\) and \(g(x)\) can share asymptotic - or end - behaviors.

Visualizing these functions helps solidify understanding of their algebraic characteristics.

Polynomial Division

Polynomial division is an algebraic technique used to simplify rational functions or determine their components. While not directly applied in this step-by-step solution, it's worth mentioning as it provides a clearer picture of how functions are composed.
In the context of rational functions, polynomial division might be used to rearrange or simplify parts of \(f(x)\) to make analysis easier. However, for determining asymptotes and end behavior, often a quick evaluation of leading terms suffices.

• Polynomial division helps transform complex expressions into more workable forms.
• It is useful for identifying slant/asymptotic behavior in certain functions.

Understanding when and how to use polynomial division is a valuable skill in handling rational functions effectively.