Question

The figure shows the plot of a function over the interval [-4, 4]. Which one of the options given CORRECTLY identifies the function? 

 

• A. \(|2 - x|\)
• B. \(|2 - |x||\)
• C. \(|2 + |x||\)
• D. \(2 - |x|\)

Hint:

Recognize graph transformations. The graph of \(|f(x)|\) is the graph of \(f(x)\) with any portion below the x-axis reflected above it. The "W" shape is a classic signature of \(|a - b|x||\) functions. Testing a few key points, especially intercepts and vertices, is a quick way to eliminate incorrect options.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The task is to identify the correct algebraic expression for the function shown in the graph. We can do this by testing key points from the graph in each of the given functional forms and by analyzing the shape of the graph, which is characteristic of absolute value functions.
Step 2: Detailed Explanation:
Let's analyze the graph for key features:
1. Symmetry: The graph is symmetric with respect to the y-axis, which means it is an even function, i.e., \(f(x) = f(-x)\). This suggests the presence of \(|x|\) in the function's definition.
2. Key Points:
- At \(x=0\), the function value is \(y=2\).
- At \(x=2\) and \(x=-2\), the function value is \(y=0\).
- At \(x=4\) and \(x=-4\), the function value appears to be \(y=2\).
3. Shape: The graph has a "W" shape, which is often created by taking the absolute value of a "V" shaped function that goes below the x-axis.
Now, let's test the options with the key points:
(A) \(f(x) = |2 - x|\):
- At \(x=-2\), \(f(-2) = |2 - (-2)| = |4| = 4\). The graph shows \(f(-2) = 0\). This option is incorrect.
(B) \(f(x) = |2 - |x||\):
- At \(x=0\), \(f(0) = |2 - |0|| = |2| = 2\). (Correct)
- At \(x=2\), \(f(2) = |2 - |2|| = |2 - 2| = |0| = 0\). (Correct)
- At \(x=-2\), \(f(-2) = |2 - |-2|| = |2 - 2| = |0| = 0\). (Correct)
- At \(x=4\), \(f(4) = |2 - |4|| = |2 - 4| = |-2| = 2\). (Correct)
- The function \(g(x) = 2 - |x|\) is an inverted "V" shape with a peak at (0, 2) and x-intercepts at \(\pm 2\). Taking the absolute value, \(f(x) = |g(x)|\), reflects the parts of the graph below the x-axis (for \(|x|>2\)) upwards, creating the "W" shape. This option matches perfectly.
(C) \(f(x) = |2 + |x||\):
- Since \(|x| \ge 0\), \(2 + |x|\) is always positive. So, \(f(x) = 2 + |x|\).
- At \(x=2\), \(f(2) = 2 + |2| = 4\). The graph shows \(f(2)=0\). This option is incorrect.
(D) \(f(x) = 2 - |x|\):
- At \(x=4\), \(f(4) = 2 - |4| = 2 - 4 = -2\). The graph shows a positive value. This option is incorrect because the graph of the function is always non-negative.
Step 3: Final Answer:
The function that correctly represents the plot is \(|2 - |x||\).
Step 4: Why This is Correct:
The function \(f(x) = |2 - |x||\) matches all the key points tested (\(x = 0, \pm 2, \pm 4\)) and its graphical transformation (an inverted V-shape reflected in the x-axis) correctly produces the "W" shape seen in the figure.