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Chapter 2: Problem 116

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if there is one, of the function's graph.

Short Answer

Expert verified
To find the horizontal asymptote of a rational function \(f(x) = \frac{p(x)}{q(x)}\), check the degrees of \(p(x)\) and \(q(x)\). If the degree of \(p(x)\) is less than the degree of \(q(x)\), the x-axis or \(y = 0\) is the horizontal asymptote. If degrees are equal, the horizontal asymptote is \(y = \frac{a}{b}\) where \(a\) and \(b\) are the leading coefficients of \(p(x)\) and \(q(x)\) respectively. If the degree of \(p(x)\) is greater, there is no horizontal asymptote.

Step by step solution

01

Understanding Rational Functions and Asymptotes

A rational function is one of the form \(f(x) = \frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomial functions and \(q(x)\) is not the zero function. An asymptote is a line that the curve approaches as it heads towards infinity.
02

Condition For a Horizontal Asymptote

A rational function will have a horizontal asymptote if the degree of the polynomial in the denominator, \(q(x)\), is greater than or equal to the degree of the polynomial in the numerator, \(p(x)\).
03

Finding the Horizontal Asymptote

If the degree of \(p(x)\) is less than the degree of \(q(x)\), the x-axis, i.e., the line \(y = 0\), is the horizontal asymptote. If the degree of \(p(x)\) is equal to the degree of \(q(x)\), the horizontal asymptote is the line \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of \(p(x)\) and \(q(x)\) respectively.
04

No Horizontal Asymptote

If the degree of the \(p(x)\) is greater than the degree of \(q(x)\), then the function does not have a horizontal asymptote.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\). b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\).

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x-1}$$

Find the horizontal asymptote, if there is one, of the graph of rational function. $$f(x)=\frac{-3 x+7}{5 x-2}$$
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