Question

What is the area of the triangle bounded by the graph of the function given by \(f(x) = |x - 1| - x\) with the coordinate axes given by \(x = 0\) and \(y = 0\)?

• A. \(1/2\)
• B. \(1/4\)
• C. \(1/2\)
• D. 1

Hint:

For piecewise absolute value functions, break into cases and graph manually to find intersection with coordinate axes.

Answer 1

The Correct Option is B

Step 1: Analyze the function 
\[ f(x) = |x - 1| - x \] Break into cases: Case 1: \(x \leq 1\) \[ f(x) = (1 - x) - x = 1 - 2x \] Case 2: \(x>1\) \[ f(x) = (x - 1) - x = -1 \] So the function is: \[ f(x) = \begin{cases} 1 - 2x & x \leq 1 
-1 & x>1 \end{cases} \]

Step 2: Area bounded by graph, x-axis, and y-axis 
We consider only \(x \in [0,1]\), where the graph is \(f(x) = 1 - 2x\) This is a line from (0,1) to (0.5,0). So triangle between (0,0), (0,1), and (0.5,0) \[ \text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 0.5 \cdot 1 = \boxed{\frac{1}{4}} \]