Problem 83 - Algebra and Trigonometry

Question

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\)

Step-by-Step Solution

Answer

The graph of \(f(x)\) can visualize the solution set of the inequality \(f(x) < 0\) by identifying where the graph is below the x-axis. The x-values in these regions compose the solution set of the inequality.

Step 1: Understanding the inequality \(f(x) < 0\)

The inequality \(f(x) < 0\) means that we want to find all values of \(x\) that lead to negative function values. In other words, where is the function \(f(x)\) under the x-axis.

Step 2: Plotting the function graph

Graph the function \(f(x)\). This could be done either by sketching the function if familiar with the shape of the function type, or by using a graphing tool. The shape could be a straight line for a first-degree polynomial, a U-shape or n-shape for a second-degree polynomial, etc., and for rational functions it could be a hyperbola.

Step 3: Identifying regions where \(f(x) < 0\)

To visualize the solution set, identify the regions where the graph is below the x-axis. This is the region where \(f(x) < 0\). The x-axis divides the plane into two parts, the upper half where \(f(x) > 0\), and the lower half where \(f(x) < 0\). The goal for this exercise is to find the x-values for when \(f(x)\) is below the x-axis.

Step 4: Interpreting the graph

The x-coordinate of the points in the region we have identified form the solution set of our inequality. In an interval notation, our solution may look something like \((-∞, a)\) or \((b, ∞)\) or can have multiple intervals like \((-∞, a) ∪ (b, ∞)\). This says that all x-values in these intervals make \(f(x) < 0\).