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Chapter 3: Problem 23

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$g(x)=\frac{x+3}{x(x+4)}$$

Short Answer

Expert verified
The function \(g(x)=\frac{x+3}{x(x+4)}\) has vertical asymptotes at \(x = 0\) and \(x = -4\). There are no holes in the function's graph.

Step by step solution

01

Find the Denominator

Firstly, we need to find the denominator of the given rational function. The denominator for the function \(g(x)=\frac{x+3}{x(x+4)}\) is \(x(x+4)\)
02

Finding values that make the denominator zero

Next, we set the denominator equal to zero and solve the equation: \[x * (x + 4) = 0\] This equation is true if \(x = 0\) or \(x = -4\). These values might result in vertical asymptotes or holes, depending on whether they are also roots of the numerator expression.
03

Checking for roots in the numerator

We check if the values of \(x\) that result in a zero denominator are also roots of the numerator. In this case, \(x = -3\) makes the numerator 0, but since neither 0 nor -4 are equal to -3, these numbers do not result in holes.
04

Defining the Asymptotes

Finally, as neither of the values \(x = 0\) or \(x = -4\) result in 0 in the numerator, they are not hole-presenting values, but they are vertical asymptotes. Thus the vertical asymptotes of the function \(g(x)\) are \(x = 0\) and \(x = -4\).

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Most popular questions from this chapter

Use a graphing utility to graph \(y-\frac{1}{x^{2}}, y-\frac{1}{x^{4}},\) and \(y-\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y-\frac{1}{x^{n}} ?\)

The rational function $$f(x)-\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\). a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$

The annual yield per lemon tree is fairly constant at 320 pounds when the number of trees per acre is 50 or fewer. For each additional tree over \(50,\) the annual yield per tree for all trees on the acre decreases by 4 pounds due to overcrowding. Find the number of trees that should be planted on an acre to produce the maximum yicld. How many pounds is the maximum yield?

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+4}{x}$$
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