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Chapter 1: Problem 89

Use a graphing utility. Graph: \(f(x)=\left|x^{2}-1\right|-|x-2|\)

Short Answer

Expert verified
To graph \(f(x)=|x^{2}-1|-|x-2|\), you should note that where the function crosses the x-axis at \(x = 1\), \(x = -1\), and \(x = 2\) and a noticeable minimum point exists at (2, 3). The graph appears as two separate V-shapes because of the combination of the two absolute value functions.

Step by step solution

01

Understanding Absolute Value Function

An absolute value function is a function that contains an algebraic expression within absolute value symbols. The output of this function is always positive, no matter what value of 'x' you input. In this case, the function is \(f(x)=|x^{2}-1|-|x-2|\), which combines two absolute values.
02

Input the Function into the Graphing Utility

The next step would be to input this function into a graphing utility. Many online platforms like Desmos or a physical graphing calculator like a TI-84 could be used to graph this function accurately.
03

Analyze the Graph

The graph should appear as a V-shape as a result of the absolute value function. Look for key points such as where the graph intersects the x and y axes. These points provide valuable information about the function, for instance, the x-intercepts indicate the solutions to the equation when \(f(x) = 0\) and the y-intercept represents the function's value when \(x = 0\).
04

Identify Key Points on the Graph

Key points to identify in this graph are the points at which the function crosses the axes. Those major points are usually where \(x = 1\), \(x = -1\), and \(x = 2\) as they make the expressions in the absolute value equal to zero. In addition, you should mark the point (2, 3), which is a minimum point in this function and distinguish the areas where the function mirrors itself because of the absolute value.

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