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Chapter 3: Problem 83

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 \(f(x)<0\)

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.
04

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 \((-鈭�, a)\) or \((b, 鈭�)\) or can have multiple intervals like \((-鈭�, a) 鈭� (b, 鈭�)\). This says that all x-values in these intervals make \(f(x) < 0\).

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Most popular questions from this chapter

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Use $$ \frac{2 x^{3}-3 x^{2}-11 x+6}{x-3}=2 x^{2}+3 x-2 $$ to factor \(2 x^{3}-3 x^{2}-11 x+6\) completely

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has vertical asymptotes given by x=-2 and x=2, a horizontal asymptote y=2, y -intercept at \frac{9}{2}, x -intercepts at -3 and 3, and y -axis symmetry.

In Exercises 104鈥�107, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) falls to the left and falls to the right.

Use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. \(f(x)=-2 x^{3}+6 x^{2}+3 x-1\)

In Exercises 94鈥�97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{4}+8 x^{3}+4 x^{2}+2$$
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