Q. which graph represents the function f(x) = |x|?

Solution — Graph of the function \(f(x)=|x|\)

Step 1 — Use the definition of absolute value

The absolute value function is defined by the piecewise rule:
\[
f(x)=|x| =
\begin{cases}
x & \text{if } x \ge 0,\\[6pt]
-x & \text{if } x < 0.
\end{cases}
\]

Step 2 — Analyze each piece separately

1. For \(x \ge 0\): the formula is \(f(x)=x\). This is a straight line with slope 1 and y-intercept 0. Points on this part include \((0,0), (1,1), (2,2),\) etc.
2. For \(x < 0\): the formula is \(f(x)=-x\). This is a straight line with slope \(-1\) and y-intercept 0 when extended. Points on this part include \((-1,1), (-2,2),\) etc.

Step 3 — Combine the two pieces

Both pieces meet at the origin \((0,0)\). For nonnegative x the graph follows the line \(y=x\); for negative x the graph follows the line \(y=-x\). Thus the complete graph is two rays meeting at the origin, forming a V shape.

Step 4 — Additional properties (useful for identifying the correct graph)

• Vertex: \((0,0)\).
• Domain: all real numbers, \((-\infty,\infty)\).
• Range: \(y \ge 0\), i.e. \([0,\infty)\).
• Symmetry: even function, symmetric about the y-axis.
• Continuity: continuous everywhere; differentiable everywhere except at \(x=0\) (corner point).

Step 5 — How to recognize the correct graph

Look for a V-shaped graph with its lowest point (vertex) at the origin, the right arm along the line \(y=x\) for \(x \ge 0\), and the left arm along the line \(y=-x\) for \(x \le 0\). The graph never goes below the x-axis.