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Chapter 3: Problem 83

Find the slant asymptote of the graph of each rational function and \(\mathbf{b}\). Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$ f(x)=\frac{x^{2}+1}{x} $$

Short Answer

Expert verified
The slant asymptote of the given rational function \(f(x) = \frac{x^{2} + 1}{x}\) is \(y=x\).

Step by step solution

01

Perform Polynomial Division

Rewrite the function as a division problem: \(x^{2}+1\) divided by \(x\). '1' can be written as \(x^{0}\) so as to have the same base 'x' to perform the division. So, the function is now: \(x^{2}/x + 1/x = x + 1/x\).
02

Finding the Slant Asymptote

Observe that for large values of 'x', \(1/x\) becomes very small and tends to zero. Thus, the asymptote is \(y = x\) as 'x' approaches 卤 infinity. We can verify this by calculating the limit of the function \(f(x)\) as 'x' approaches 卤 infinity. As x goes to 卤 infinity, \(f(x)\) goes to \(x+0\), which simplifies to \(x\). Therefore, the slant asymptote of the graph of the rational function is \(y=x\).
03

Graphing the rational function with the slant Asymptote

Draw a graph with 'x' on the horizontal axis and 'f(x)' on the vertical axis. Draw the line \(y=x\) as the slant asymptote. Now, draw the graph of the function \(f(x)\). For values of 'x'tending towards 卤 infinity verify that the graph approaches the line \(y=x\) an asymptotic behavior in both the positive and negative directions. This confirms that \(y=x\) is indeed the slant asymptote of the given function.

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Key Concepts

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

Rational Functions
Rational functions are expressions where a polynomial is divided by another polynomial. In the given exercise, the rational function is expressed as \( f(x) = \frac{x^2 + 1}{x} \).
This means we have a polynomial \( x^2 + 1 \) in the numerator and simply \( x \) in the denominator. Rational functions can have various types of asymptotes, such as vertical, horizontal, or slant (oblique).
Understanding the behavior of rational functions at infinity helps us to determine these asymptotes.
  • Vertical asymptotes occur when the denominator equals zero and the function becomes undefined.
  • Horizontal asymptotes are established by the degrees of the polynomials in the numerator and the denominator.
  • Slant asymptotes, like the one in the exercise, appear when the degree of the numerator is exactly one more than the degree of the denominator.
Knowing these allows us to visualize and graph rational functions better.
Polynomial Division
To find slant asymptotes, polynomial division is a crucial tool. It involves dividing the polynomial in the numerator by the polynomial in the denominator.
In this problem, we divide \( x^2 + 1 \) by \( x \) using synthetic or long division. This results in \( x + \frac{1}{x} \), simplifying to \( x \) as \( x \) becomes very large or small.
Here鈥檚 a small guide to performing polynomial division:
  • Align coefficients of like terms in both polynomials.
  • Divide the first term of the numerator by the first term of the denominator.
  • Multiply the divisor by that result and subtract it from the dividend.
  • Bring down the next term of the dividend if necessary and repeat the process.
The remainder, in this case \( \frac{1}{x} \), lessens as \( x \) extends towards infinity, guiding us to the slant asymptote \( y = x \).
Graphing Asymptotes
Graphing asymptotes, especially slant ones, provides visual insights into the behavior of rational functions. A slant asymptote isn鈥檛 horizontal or vertical; instead, it's a diagonal line on the graph.
Here's how to plot them:
  • Draw the asymptote line, in this case \( y = x \), to guide the graph formation.
  • Make sure the graph follows the direction towards \( y = x \) as \( x \) moves towards infinity.
  • It approaches the asymptote but never touches or crosses it at extreme values of \( x \).
In the given function \( f(x) = \frac{x^2 + 1}{x} \), you will see that as \( x \) becomes large in the positive or negative sense, the graph approaches this asymptote. This behavior is important for predicting function behavior at extremes, aiding in drawing accurate sketches of graph behavior.

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