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Chapter 3: Problem 94

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

Short Answer

Expert verified
The simplified function is \(\frac{x - 1}{x + 1}\). Sketch the graph by choosing values of \(x\), calculating \(f(x)\), and plotting these points and taking note of where the function is undefined.

Step by step solution

01

Simplify the Numerator

First, consider the numerator of the given expression. To simplify, find a common denominator for \(x\) and \(\frac{1}{x}\). The common denominator is \(x\). So the numerator becomes \(\frac{x^2}{x} - \(\frac{1}{x}*\frac{x}{x}\). This simplifies to \(x - 1\).
02

Simplify the Denominator

Next, consider the denominator. Again, find a common denominator for \(x\) and \(\frac{1}{x}\). This denominator is also \(x\). So the denominator becomes \(\frac{x^2}{x} + \(\frac{1}{x}*\frac{x}{x}\). This simplifies to \(x + 1\).
03

State the Simplified Function

The simplified function becomes \(\frac{x - 1}{x + 1}\). This is the function that we can graph.
04

Graph the Function

With the simplified function \(\frac{x - 1}{x + 1}\), create the graph. To do this, select values for \(x\) and compute the corresponding \(f(x)\) values. Then plot these points on a Cartesian plan. Remember to mark on the graph where the function is undefined (where the denominator equals zero).

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