To find the area bounded by the curve \( f(x) = \max(|x| - 1, 1 - |x|) \) with the x-axis from \( x = -1 \) to \( x = 1 \), we will follow these steps:
### Step 1: Analyze the function
First, we need to analyze the function \( f(x) = \max(|x| - 1, 1 - |x|) \).
- The expression \( |x| - 1 \) is negative for \( -1 < x < 1 \) and equals 0 at \( x = -1 \) and \( x = 1 \).
- The expression \( 1 - |x| \) is positive for \( -1 < x < 1 \) and equals 0 at \( x = -1 \) and \( x = 1 \).
### Step 2: Determine the intervals
Next, we will find the points where the two expressions are equal:
\[
|x| - 1 = 1 - |x|
\]
- For \( x \geq 0 \): \( x - 1 = 1 - x \) leads to \( 2x = 2 \) or \( x = 1 \).
- For \( x < 0 \): \( -x - 1 = 1 + x \) leads to \( -2x = 2 \) or \( x = -1 \).
Thus, the expressions are equal at \( x = -1 \) and \( x = 1 \).
### Step 3: Evaluate the function in the intervals
Now we evaluate \( f(x) \) in the intervals:
- For \( -1 \leq x < 0 \):
\[
f(x) = 1 - |x| = 1 + x
\]
- For \( 0 \leq x \leq 1 \):
\[
f(x) = 1 - |x| = 1 - x
\]
### Step 4: Sketch the graph
Now we can sketch the graph of \( f(x) \):
- From \( x = -1 \) to \( x = 0 \), the line \( f(x) = 1 + x \) goes from \( ( -1, 0 ) \) to \( ( 0, 1 ) \).
- From \( x = 0 \) to \( x = 1 \), the line \( f(x) = 1 - x \) goes from \( ( 0, 1 ) \) to \( ( 1, 0 ) \).
### Step 5: Calculate the area
The area under the curve from \( x = -1 \) to \( x = 1 \) can be calculated as the area of the triangle formed between the points \( (-1, 0) \), \( (0, 1) \), and \( (1, 0) \).
- The base of the triangle is the distance from \( -1 \) to \( 1 \), which is \( 2 \) units.
- The height of the triangle is the maximum value of \( f(x) \), which is \( 1 \) unit.
Using the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 1 = 1
\]
### Final Answer
The area bounded by the curve and the x-axis from \( x = -1 \) to \( x = 1 \) is \( 1 \) square unit.
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