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Chapter 3: Problem 62

Find the oblique asymptote and sketch the graph of each rational function. $$f(x)=\frac{x^{2}-1}{x}$$

Short Answer

Expert verified
The oblique asymptote is \(y = x\).

Step by step solution

01

Perform Polynomial Long Division

Divide the numerator by the denominator using polynomial long division to express the function in the form of a linear term plus a remainder. Divide \(x^2 - 1\) by \(x\): \(\frac{x^2 - 1}{x} = x - \frac{1}{x}\)
02

Identify the Oblique Asymptote

The oblique asymptote can be identified as the quotient obtained from the polynomial long division, without the remainder. So, the oblique asymptote is given by \(y = x\).
03

Analyze the Behavior of the Function

Examine how the function behaves as \(x\) approaches positive and negative infinity. As \(x\) becomes very large or very small, the term \(-\frac{1}{x}\) will approach 0, and the function will resemble \(f(x) \approx x\).
04

Sketch the Graph

Draw the graph of \(f(x) = \frac{x^2 - 1}{x}\). Plot the oblique asymptote line \(y = x\). Then, plot key points and note that the graph will approach the line \(y = x\) as \(x\) tends to positive or negative infinity. Also, consider any intercepts and asymptotic behavior at \(x = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Rational Functions
Graphing rational functions involves a few steps. Let's break it down using \(f(x) = \frac{x^2 - 1}{x}\).

First, identify the asymptotes. We found that the oblique asymptote is \(y = x\).

Next, plot key points. Find the y-intercept by setting \(x = 0\). Here, that point does not exist because the function is undefined at \(x = 0\) (division by zero).

Now, to sketch the graph:
  • Draw the oblique asymptote line \(y = x\).
  • Look for intercepts: the x-intercepts are where \(f(x) = 0\). For our function, set the numerator \(x^2 - 1 = 0\). This gives \(x = \pm 1\).


Finally, understand the end behavior: as \(x\) goes to 卤鈭�, the term \(-\frac{1}{x}\) tends to 0, making \(f(x)\) almost equal to \(y = x\). This means the graph will approach the line \(y = x\) from both directions.

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