Problem 94 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(x - 1\).

Step 2: 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 \(x + 1\).

Step 3: State the Simplified Function

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

Step 4: 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).