Problem 89 - 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. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

Step-by-Step Solution

Answer

The simplified function is \(f(x) = \frac{1}{2x}\). The graph of the function is a hyperbola in the first and third quadrants, with the x-axis as its asymptote.

Step 1: Simplification

First, simplify the given expression. Factorize the component expressions as follows: \(x^{2}-4 = (x-2)(x+2)\) and \(x^{2}+4x+4= (x+2)^{2}\). The simplified function becomes \(f(x) = \frac{5 x^{2}}{(x-2)(x+2)} \cdot \frac{(x+2)^{2}}{10x^{3}}\). Notice that the \(x+2\) terms cancel out, leaving us with the equation \(f(x) = \frac{1}{2}\cdot\frac{1}{x}\), after dividing terms and simplifying further.

Step 2: Graphing the Function

To plot the graph, first make a small table with values for \(x\) and corresponding values for \(f(x)\). You can start with -3, -2, -1, 1, 2, 3 for \(x\). Note that as \(x\) approaches 0 from either side, \(f(x)\) will approach positive or negative infinity. This is the behavior of hyperbolic functions. Also, remember that the function is undefined at \(x = 0\), hence, the graph will not touch the x-axis at x = 0.

Step 3: Finished Graph

When you plot these points and connect them, you will see that the graph is a hyperbola. The graph will be in the first and third quadrants only, because the function is only defined for \(x \neq 0\). This finalizes our graphing process for the function \(f(x)\).