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Chapter 2: 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 equation for \(f\) is \(f(x) = \frac{x^2-1}{x^2+1}\). It is symmetrical about the origin and the graph approaches, but never reaches, the line \(y=1\). There are no restrictions on the domain.

Step by step solution

01

- Simplify the equation

To simplify the expression \(\frac{x-\frac{1}{x}}{x+\frac{1}{x}}\), start by multiplying both the numerator and the denominator by \(x\). This will give \(\frac{x^2-1}{x^2+1}\). Hence the equation for \(f\) is \(f(x) = \frac{x^2-1}{x^2+1}\).
02

- Graph the function

Plotting the graph of the function, you will first note that the denominator \(x^2 + 1\) is always positive (for every real \(x\)), so there are no restrictions on the domain of the function. The function approaches 1 as \(x\) approaches plus or minus infinity, producing a horizontal asymptote at \(y=1\). Thus, the graph starts from \(-1\), slowly rises, approaches but never reaches the horizontal line \(y=1\), and is symmetric with respect to the origin.

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