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Chapter 9: Problem 53

Create a function whose graph has the given characteristics. (There are many correct answers.) Vertical asymptote: \(x=5\) Horizontal asymptote: \(y=0\)

Short Answer

Expert verified
The function \(f(x) = \frac{1}{x-5}\) has a vertical asymptote at \(x=5\) and a horizontal asymptote at \(y=0\) as required.

Step by step solution

01

Formulate the Function

A simple way to formulate a function with these asymptotes is to make the function a fraction where \(x-5\) is in the denominator and a constant is in the numerator. This will create the vertical asymptote at \(x=5\). The function will approach 0 as \(x\) approaches positive or negative infinity due to the degree of the denominator being greater than the numerator. An example function with these properties can be \(f(x) = \frac{1}{x-5}\).
02

Check the Vertical Asymptote

Let's check that this function has a vertical asymptote at \(x=5\). As \(x\) approaches 5 from the left or right, the function will approach negative or positive infinity respectively. This confirms the presence of a vertical asymptote at \(x=5\).
03

Check the Horizontal Asymptote

It's also necessary to confirm that this function has a horizontal asymptote at \(y=0\). As \(x\) approaches either positive or negative infinity, the value of the function will approach 0. Thus, the function does have a horizontal asymptote at \(y=0\).

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Most popular questions from this chapter

The cost and revenue functions for a product are \(C=34.5 x+15,000\) and \(R=69.9 x\) (a) Find the average profit function \(\bar{P}=(R-C) / x\). (b) Find the average profits when \(x\) is \(1000,10,000\), and 100,000 (c) What is the limit of the average profit function as \(x\) approaches infinity? Explain your reasoning.

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Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=x+\frac{32}{x^{2}}\)

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