Problem 526 - Precalculus

Question

For the following exercises, use a calculator to graph \(f(x).\) Use the graph to solve \(f(x) > 0\) $$f(x)=\frac{2}{(x-1)(x+2)}$$

Step-by-Step Solution

Answer

\(-2 < x < 1\).

Step 1: Analyze the Function

The given function is \(f(x) = \frac{2}{(x-1)(x+2)}\). This is a rational function with potential discontinuities at \(x = 1\) and \(x = -2\), where the denominator is zero.

Step 2: Identify Discontinuities

Determine where \((x-1)(x+2) = 0\). Solving \((x-1) = 0\) gives \(x = 1\), and \((x+2) = 0\) gives \(x = -2\). These are vertical asymptotes of the graph.

Step 3: Graph the Function

Using a calculator, graph the function. Notice that the function approaches vertical asymptotes at \(x = -2\) and \(x = 1\).

Step 4: Determine Where \(f(x) > 0\)

Examine the graph to identify where the curve is above the x-axis. From the graph, \(f(x)\) is greater than zero when \(-2 < x < 1\).

Step 5: Conclude the Solution

The solution to \(f(x) > 0\) is the interval \(-2 < x < 1\), based on the graph analysis.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are crucial in understanding the behavior of rational functions. They occur at values of \(x\) where the function becomes undefined, typically when the denominator is equal to zero.
In the function given, \(f(x) = \frac{2}{(x-1)(x+2)}\), identifying the points where the denominator equals zero shows us where the vertical asymptotes are located. Solving the equations \((x-1)=0\) and \((x+2)=0\) gives us the vertical asymptotes at \(x = 1\) and \(x = -2\).
Understanding vertical asymptotes helps in graphing, as they indicate where the graph will have breaks and approach infinitely. They essentially split the graph into sections where the behavior might differ – either increasing or decreasing sharply. Spotting these helps when exploring where a function may exist above or below the x-axis.

Graphing Calculator

Using a graphing calculator simplifies the visualization of complex functions like rational ones. Graphing \(f(x) = \frac{2}{(x-1)(x+2)}\) allows us to quickly and accurately identify features of the function.
By inputting the function, the calculator displays the graph, showing how the function behaves as \(x\) approaches the vertical asymptotes at \(x=1\) and \(x=-2\). This technology saves time and provides a clear picture of the function's overall shape and behavior.

• See how the curve approaches the asymptotes without actually touching them
• Identify when and where the function is positive or negative

The graph's visual output is invaluable for examining periods of positive or negative value across different intervals of \(x\). This makes solving inequalities much more intuitive.

Solutions to Inequalities

When solving inequalities such as \(f(x) > 0\), it's important to determine where the function's curve lies above the x-axis. For the function \(f(x) = \frac{2}{(x-1)(x+2)}\), the question is about identifying intervals where \(f(x)\) is positive.
After graphing the function, you can clearly see where the graph is above the x-axis. In this example, the graph shows that \(f(x) > 0\) when \(-2 < x < 1\).
Analyzing the solution:

• The graph begins to rise above the x-axis just right of \(x = -2\)
• The curve then crosses back below the x-axis right at \(x = 1\)

These insights about where a function is above or below the axis provide definite boundaries for your solution, which in this case, ends up being a clear interval.