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Chapter 2: Problem 91

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x}{2 x+6}-\frac{9}{x^{2}-9}$$

Short Answer

Expert verified
The simplified form of the given function is \(f(x) =\frac{x^{2} -3x -18}{2(x-3)(x+3)}\). The graph of this function is a hyperbola with vertical asymptotes at \(x=3\) and \(x=-3\).

Step by step solution

01

Factor the denominators

To lump these two fractions into one, we need to find a common denominator. That often involves factoring. The denominators \(2x+6\) and \(x^{2}-9\) factor as \(2(x+3)\) and \((x-3)(x+3)\) respectively. Notice that both have \(x+3\) in common, that's a good sign.
02

Combine into a single subtracted fraction

Now, we can rewrite each fraction in terms of the common denominator \((x-3)(x+3)\). Specifically, \(\frac{x}{2(x+3)}\) becomes \(\frac{x(x-3)}{2(x-3)(x+3)}\) and \(\frac{9}{(x-3)(x+3)}\) stays as it is. We can then subtract them: \(\frac{x(x-3)}{2(x-3)(x+3)} - \frac{9}{(x-3)(x+3)}\) results in \(\frac{x(x-3)-18}{2(x-3)(x+3)}\).
03

Simplify the numerator

The numerator \(x(x-3)-18\) simplifies to \(x^{2} -3x -18\). Substitute this back into the fraction from step 2.
04

Write f(x)

Finally, we write the simplified expression for f as: \(f(x) =\frac{x^{2} -3x -18}{2(x-3)(x+3)}\)
05

Graph f(x)

Plot the graph of the function \(f(x) =\frac{x^{2} -3x -18}{2(x-3)(x+3)}\). Be sure to include any relevant points, intercepts and asymptotes. Because of the x term in the denominator, there would be vertical asymptotes at \(x=3\) and \(x=-3\), and because of the \(x^{2}\) term in the numerator, the graph will be a hyperbola.

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Most popular questions from this chapter

If you are given the equation of a rational function, explain how to find the vertical asymptotes, if any, of the function's graph.

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 \(y=\frac{1}{x^{n}} ?\)

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?

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 \(x=3\).

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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