Q. Which graph represents the function \(p(x) = |x – 1|\)?

Problem

Which graph represents the function \(p(x)=|x-1|\)?

Step-by-step detailed explanation

1. Understand the absolute value expression.

   The absolute value of an expression takes the expression if it is nonnegative and the negative of the expression if it is negative. For the expression \(x-1\), split into two cases based on the sign of \(x-1\).

2. Write the function as a piecewise linear function.

   Use the definition of absolute value to obtain a piecewise form.

   \[
   p(x)=|x-1|=\begin{cases}
   x-1 & \text{if } x-1\ge 0,\\[6pt]
   -(x-1) & \text{if } x-1<0. \end{cases} \]

   Simplify the conditions and the second piece:

   \[
   p(x)=\begin{cases}
   x-1 & \text{if } x\ge 1,\\[4pt]
   -x+1 & \text{if } x<1. \end{cases} \]

3. Locate the vertex (the meeting point of the two pieces).

   Both pieces meet at \(x=1\). Evaluate \(p(1)\) using either piece:

   \(p(1)=|1-1|=0\).

   The vertex is the point \((1,0)\). This is the bottom point of the V-shaped graph.

4. Determine slopes of the two rays.

   For \(x\ge 1\) the function is the line \(y=x-1\), which has slope \(1\).

   For \(x<1\) the function is the line \(y=-x+1\), which has slope \(-1\).

   So the right arm of the V rises with slope 1, and the left arm rises (to the left) with slope -1.

5. Find intercepts and a few points to plot.

   x-intercept: set \(p(x)=0\) gives \(|x-1|=0\) so \(x=1\). Point \((1,0)\).

   y-intercept: evaluate \(p(0)=|0-1|=1\). Point \((0,1)\).

   Additional points: \(p(2)=|2-1|=1\) gives \((2,1)\); \(p(-1)=|-1-1|=2\) gives \((-1,2)\).

6. Describe the complete graph from these facts.

   The graph is a V-shape with vertex at \((1,0)\). For \(x\ge 1\) the graph follows the straight line \(y=x-1\). For \(x<1\) the graph follows the straight line \(y=-x+1\). The graph is symmetric about the vertical line \(x=1\).

7. State domain and range.

   Domain: all real numbers, written \((-\infty,\infty)\).

   Range: all real numbers greater than or equal to zero, written \([0,\infty)\).

Conclusion: Which graph represents p(x)=|x-1|?

The correct graph is the V-shaped graph with vertex at \((1,0)\), left arm a line of slope \(-1 (the line \(y=-x+1\)) for \(x<1\), and right arm a line of slope \(1 (the line \(y=x-1\)) for \(x\ge 1\). It crosses the y-axis at \((0,1)\) and has range \([0,\infty)\).