Problem 72 - College Algebra with Corequisite Support

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

\(f(x) > 0\) when \(x \in (-\infty, -2) \cup (1, \infty)\).

Step 1: Identify the Function

The given function is \(f(x) = \frac{2}{(x-1)(x+2)}\). This is a rational function with the numerator as 2 and the denominator as the product of \((x-1)\) and \((x+2)\).

Step 2: Find the Vertical Asymptotes

The vertical asymptotes occur where the denominator is zero. Solve \((x-1)(x+2) = 0\) to find the vertical asymptotes. \(x = 1\) and \(x = -2\) are the vertical asymptotes.

Step 3: Graph the Function

Using a graphing calculator, input \(f(x) = \frac{2}{(x-1)(x+2)}\). Observe the behavior near the asymptotes at \(x = 1\) and \(x = -2\). The graph should have branches approaching the asymptotes and possibly change between positive and negative values.

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

Look at the graph and identify the regions where the curve is above the x-axis. This occurs where \(f(x) > 0\). Check intervals between the vertical asymptotes and beyond.

Step 5: Solve Inequality

The graph is positive in the region \((-\infty, -2)\) and \((1, \infty)\). Therefore, the solution to \(f(x) > 0\) is \((-\infty, -2) \cup (1, \infty)\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never actually touches or crosses. In rational functions like the one given, they occur at points where the denominator equals zero, causing the function to become undefined. For the function \(f(x) = \frac{2}{(x-1)(x+2)}\), we find the vertical asymptotes by setting the denominator equal to zero: \((x-1)(x+2) = 0\). Solving this equation gives us the vertical asymptotes at \(x = 1\) and \(x = -2\).
These asymptotes serve as boundaries on the graph where the function will shoot off towards positive or negative infinity. Visually, the graph of the function will get closer and closer to these vertical lines without touching them, creating a distinct separation in the behavior of the function on either side of the asymptote. Understanding vertical asymptotes is crucial when analyzing the behavior of rational functions, as they impact the limits and continuity of the function.

Graphing Calculators

A graphing calculator is a handy tool when you want to visualize rational functions like \(f(x) = \frac{2}{(x-1)(x+2)}\). These calculators can quickly plot the graph of a function, allowing you to see important features such as asymptotes, intersections, and the behavior of the curve.
To graph the function, you input the expression into the calculator. As it plots the graph, you can observe how the curve behaves near the asymptotes \(x = 1\) and \(x = -2\). The graph will branch out approaching these lines without crossing them, and you may notice that the curve stretches far away from the x-axis vertically when it is near an asymptote.

• Zooming in and out on the graph can give you a better perspective of the function's behavior.
• Using the calculator's features, you can trace along the graph to find specific points and values.

Graphing calculators are incredibly useful for solving inequalities and understanding complex functions as they provide a visual representation that can make these abstract concepts much more tangible.

Inequalities

Inequalities describe a range of values that satisfy a certain condition, such as \(f(x) > 0\). This means finding where the graph of the function lies above the x-axis, indicating positive values.
Analyzing the graph of \(f(x) = \frac{2}{(x-1)(x+2)}\), you can determine the intervals where the function is greater than zero. From our earlier steps with the vertical asymptotes, we found the critical points \(x = 1\) and \(x = -2\).
To determine where \(f(x) > 0\):

• The function is positive between the intervals \((-\infty, -2)\) and \((1, \infty)\). In these regions, the graph remains above the x-axis.
• This is visually confirmed by noting where the graph does not dip below the horizontal line representing our zero marker – the x-axis itself.

By comprehending inequalities within the context of rational functions, you can identify specific intervals where solutions like \(f(x) > 0\) apply, helping solve practical mathematical problems.