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Chapter 2: 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
After simplification, the equation for \(f\) is \(f(x) = \frac{x -3 - \frac{10}{x}}{x +1 -\frac{2}{x}}\), for \(x \neq 0\). To graph the function, create a table of values for a range of x-values and calculate the corresponding f(x) values.

Step by step solution

01

Simplification

To simplify the expression, let鈥檚 get rid of the complex fractions by multiplying the numerator and denominator by \((x+2)(x-2)\). This is the least common multiple (LCM) of the denominators of the fractions in the expression. Thus we have:\n\((x+2)(x-2)\Bigl(1-\frac{3}{x+2}\Bigr) / (x+2)(x-2)\Bigl(1+\frac{1}{x-2}\Bigr) = \frac{(x+2)(x-2) - (x+2)*3}{(x+2)(x-2) + (x+2)}\)
02

Expand and Simplify

Expanding and simplifying the numerator and denominator in the above expression yields:\n\(\frac{x^{2}-4 -3x-6}{x^{2} - 4 + x + 2} = \frac{x^{2}-3x - 10}{x^{2} +x -2}\)
03

Simplify Further

We continue to simplify the equation, by dividing each term in the numerator and the denominator by x, only if x is not equal to 0. Therefore, we get: \n\(f(x) = \frac{x -3 - \frac{10}{x}}{x +1 -\frac{2}{x}}\)
04

Graphing the Function

To graph this function, create a table of values for a series of x-values and calculate the corresponding f(x) values. The graph of the function represents these ordered pairs (x, f(x)). Be sure to include the points of discontinuity which occur when x is -2 or 2 since these x-values would make the original rational function undefined.

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Most popular questions from this chapter

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