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Chapter 2: Problem 9

Sketch the graph of the equation \(y=|x|-|x-1|\)

Short Answer

Expert verified
The graph of the given function is a straight horizontal line at \(y=1\).

Step by step solution

01

Identify Change Points

Before starting to sketch the graph, identify the critical points of the absolute value expressions. These will be the points where the expression inside the absolute value sign equals to zero. For the expression \(|x|\), the change point is \(x=0\) and for the expression \(|x-1|\) it is \(x=1\)
02

Form Piecewise Function

Form the piecewise function from the given equation, based on the change points identified in step 1. That is: For \(x < 0\), \(y = -x - (-x+1)\); which simplifies to \(y=1\). For \(0 鈮� x < 1\), \(y = x - (x-1)\); simplifying to \(y = 1\). And for \(x 鈮� 1\), \(y=x-(x-1)\), which simplifies to \(y=1\). Thus, the piecewise function for all \(x\) becomes \(y=1\).
03

Draw the Graph

Based on the piecewise function derived in the previous step, it's clear that the value of \(y\) remains constant for all \(x\). Thus, the graph is a straight horizontal line \(y=1\) that spans the entire x-axis.

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