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Chapter 2: Problem 78

Determine whether the statement is true or false. Justify your answer. The graph of a rational function can never cross one of its asymptotes.

Short Answer

Expert verified
The statement is partially true and partially false. If we are referring to vertical asymptotes, the graph of a rational function cannot cross them. However, if we are referring to horizontal or oblique asymptotes, the graph of a rational function can indeed cross them.

Step by step solution

01

Defining Rational Function

A Rational function is any function that can be defined by a rational fraction, i.e., an algebraic fraction such that both the numerator and the denominator are polynomials. The graphs of rational functions can have various shapes, including one or more vertical asymptotes, or slant (oblique) asymptotes.
02

Understanding The Concept Of Asymptote

An asymptote is a line that a graph approaches without touching. For a rational function, there are vertical and horizontal asymptotes, determined by the degree, or highest power of x, in the numerator and denominator. Graphs tend to go to infinity or negative infinity (without reaching it) at vertical asymptotes and they approach a certain value (without reaching it) at horizontal asymptotes as x goes to positive or negative infinity.
03

Determining If A Graph Of A Rational Function Can Cross Its Asymptote

Graphs of rational functions can approach their asymptotes without touching them, which means they cannot cross their asymptotes. However, there's a catch - while vertical asymptotes cannot be crossed, horizontal or oblique asymptotes can be crossed. But the question specifically mentions 'one of its asymptotes' without specifying the type, making the answer more nuanced 鈥� if it's a vertical asymptote, the graph can't cross it; if it's a horizontal or oblique asymptote, the graph can cross it.

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(z)=z^{2}-2 z+2$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

The revenue and cost equations for a product are \(R=x(50-0.0002 x)\) and \(C=12 x+150,000,\) where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold. How many units must be sold to obtain a profit of at least \(\$ 1,650,000 ?\) What is the price per unit?

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=2 x^{3}-3 x^{2}-3$$
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