The area (in sq. units) bounded by the curve `f(x)=max(|x|-1, 1-|x|)` with the x- axis from `x=-1 " to " x =1` is

Answer

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 can follow these steps:

### Step 1: Understand the functions involved
The function \( f(x) \) is defined as the maximum of two functions: \( |x| - 1 \) and \( 1 - |x| \). We need to analyze these two functions to determine where each is greater.

### Step 2: Analyze \( |x| - 1 \)
- For \( x < -1 \): \( |x| - 1 = -x - 1 \) (which is negative).
- For \( -1 \leq x < 1 \): \( |x| - 1 = 0 \) when \( x = -1 \) and \( x = 1 \); it is negative between these points.

### Step 3: Analyze \( 1 - |x| \)
- For \( x < -1 \): \( 1 - |x| = 1 + x \) (which is also negative).
- For \( -1 \leq x < 1 \): \( 1 - |x| \) is positive and decreases from 2 to 0 as \( x \) moves from -1 to 1.

### Step 4: Determine the intervals
- From the analysis, we find that:
- For \( x = -1 \): \( f(-1) = \max(0, 2) = 2 \)
- For \( x = 0 \): \( f(0) = \max(-1, 1) = 1 \)
- For \( x = 1 \): \( f(1) = \max(0, 0) = 0 \)

### Step 5: Sketch the graph
The graph of \( f(x) \) will be:
- From \( x = -1 \) to \( x = 0 \), \( f(x) = 1 - |x| \) which is a straight line from \( ( -1, 2) \) to \( (0, 1) \).
- From \( x = 0 \) to \( x = 1 \), \( f(x) = 1 - |x| \) which is a straight line from \( (0, 1) \) to \( (1, 0) \).

### Step 6: Calculate the area
The area under the curve from \( x = -1 \) to \( x = 1 \) can be calculated as the area of the triangle formed:
- The base of the triangle is from \( x = -1 \) to \( x = 1 \) which is 2 units.
- The height of the triangle is from the x-axis to the point \( (0, 1) \) which is 1 unit.

Using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 1 = 1 \text{ square unit} \]

### Final Answer
The area bounded by the curve \( f(x) \) and the x-axis from \( x = -1 \) to \( x = 1 \) is **1 square unit**.