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Chapter 3: Problem 93

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. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

Short Answer

Expert verified
The simplified equation for function \(f\) is \(f(x) = \frac{x^2-3x+2}{x^2+4x-4}\). A graph of the function would display its behavior based on this equation

Step by step solution

01

Simplify the Complex Fraction

Begin by simplifying the complex fraction. To do this, multiply every term by the common denominator, which is \((x+2)(x-2)\). This yields: \[f(x) = \frac{(x+2)(x-2)-3(x-2)}{(x+2)(x-2)+(x+2)} \]
02

Simplify the Numerator and Denominator

Next, distribute the terms in the numerator and denominator, simplify and collect like terms. This results in:\[f(x) = \frac{x^2-4-3x+6}{x^2+4x-4}\]and further simplifies to\[f(x) = \frac{x^2-3x+2}{x^2+4x-4}\]
03

Graphing the Function

Now that the function has been obtained, plot the function by setting up a graph and determine its behavior based on the function's equation. Note: the graphing step requires numerical ability and understanding of how the function's equation affects its graphical representation. Therefore, this step is better executed using graphing tool or technology.

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Most popular questions from this chapter

Write a rational inequality whose solution set is \((-\infty,-4) \cup[3, \infty)\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Exercises will help you prepare for the material covered in the next section. Factor: \(x^{3}+3 x^{2}-x-3\)

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. $$\frac{x}{2 x+6}-\frac{9}{x^{2}-9}$$

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. $$\frac{x-\frac{1}{x}}{x+\frac{1}{x}}$$
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