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Chapter 3: Problem 29

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$f(x)=\frac{x^{2}-9}{x-3}$$

Short Answer

Expert verified
The graph of the function \(f(x)=\frac{x^{2}-9}{x-3}\) has a hole at \(x=3\) and no vertical asymptotes.

Step by step solution

01

Identify the denominator and find where it equals zero

The denominator of the function is \(x-3\). We set this equal to zero to find out where the function is undefined. So, \(x-3=0\), which implies \(x=3\).
02

Consider where the numerator equals zero

The numerator of the function is \(x^{2}-9\). We set this equal to zero to find out where it crosses the x-axis. So, \(x^{2}-9=0\). This is equivalent to \((x-3)(x+3)=0\), which tells us \(x=3\) or \(x=-3\). Note that the value \(x=3\) was also found in step 1.
03

Making conclusions about holes and asymptotes

Since \(x=3\) makes both the numerator and the denominator equal to zero, this value corresponds to a hole in the graph of the function. For \(x=-3\), it only makes the numerator zero, not the denominator, so this does not introduce a vertical asymptote nor a hole.

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Most popular questions from this chapter

Write a polynomial inequality whose solution set is \([-3,5]\)

Each group member should consult an almanac, newspaper, magaxine, or the Internet to find data that initially increase and then decrease, or vice versa, and therefore can be modeled by a quadratic function. Group members should select the two sets of data that are most interesting and relevant. For each data set selected, a. Use the quadratic regression feature of a graphing utility to find the quadratic function that best fits the data. b. Use the equation of the quadratic function to make a prediction from the data. What circumstances might affect the ac acy of your prediction? c. Use the equation of the quadratic function to write and solve a problem involving maximizing or minimizing the function.

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What is a polynomial incquality?

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-0.25 x^{2}+40 x$$
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