Q. 9 Which of the following functions describe the graph shown in the below figure?

• (A) y = ||x| + 1| – 2
• (B) y = ||x| – 1| – 1
• (C) y = ||x| + 1| – 1
• (D) y = ||x – 1| – 1|

Question Overview

The given graph shows a piecewise linear curve formed using absolute value functions.
It is symmetric about the y-axis and has a distinct W-shaped structure,
which strongly indicates a nested absolute value function.

Options Given

• (A) y = ||x| + 1| − 2
• (B) y = ||x| − 1| − 1
• (C) y = ||x| + 1| − 1
• (D) y = ||x − 1| − 1|

Key Observations from the Graph

• The graph is symmetric about the y-axis
• The graph has two minimum points at x = ±1, y = −1
• The graph passes through the origin (0, 0)
• The graph has a W-shape, typical of expressions like y = ||x| − a| − b

These characteristics narrow the answer significantly.

Correct Answer

✅ Option (B):
y = ||x| − 1| − 1

Why Option (B) Is Correct

Let’s analyze the function:

y = ||x| − 1| − 1

Step-by-Step Behavior

• |x| creates symmetry about the y-axis
• Subtracting 1 shifts the graph downward
• Taking absolute value again creates the W-shape
• Final subtraction of 1 shifts the entire graph further downward

Key Points Produced by This Function

These points perfectly match the graph, including:

• Minima at (±1, −1)
• Symmetry about the y-axis
• Passing through the origin

✔️ Hence, Option (B) is the correct answer.

Why the Other Options Are Incorrect

❌ Option (A): y = ||x| + 1| − 2

• Shifts the graph left and downward
• Minimum occurs at x = 0, not at x = ±1
• Does not produce the observed W-shape

❌ Does not match the graph

❌ Option (C): y = ||x| + 1| − 1

• Entire graph is shifted upward
• No minimum at y = −1
• Graph never touches the x-axis as shown

❌ Incorrect vertical positioning

❌ Option (D): y = ||x − 1| − 1|

• Graph is not symmetric about the y-axis
• Shifted toward the positive x-axis
• Asymmetry contradicts the given figure

❌ Fails symmetry test

Final Conclusion

✔️ The only function that:

• Produces a W-shaped graph
• Is symmetric about the y-axis
• Has minima at x = ±1
• Matches all plotted points

✅ Correct Answer: Option (B)
y = ||x| − 1| − 1