Consider the fractions:
`f (x) = |x|-1 and g (x) =1- |x|.`
(a) Find their graphs and shade the closed region between them
(b) Find the area of their shaded region.

Answer

To solve the problem step by step, we will first find the graphs of the functions \( f(x) = |x| - 1 \) and \( g(x) = 1 - |x| \), then we will shade the region between them and finally calculate the area of the shaded region.

### Step 1: Graphing \( f(x) = |x| - 1 \)

1. **Understanding the function**: The function \( f(x) = |x| - 1 \) represents the absolute value function shifted down by 1 unit.
2. **Finding key points**:
- At \( x = 0 \): \( f(0) = |0| - 1 = -1 \) (Point: \( (0, -1) \))
- At \( x = 1 \): \( f(1) = |1| - 1 = 0 \) (Point: \( (1, 0) \))
- At \( x = -1 \): \( f(-1) = |-1| - 1 = 0 \) (Point: \( (-1, 0) \))
3. **Graphing**: The graph will have a V-shape, opening upwards, with the vertex at \( (0, -1) \) and intersecting the x-axis at \( (1, 0) \) and \( (-1, 0) \).

### Step 2: Graphing \( g(x) = 1 - |x| \)

1. **Understanding the function**: The function \( g(x) = 1 - |x| \) represents the absolute value function reflected over the x-axis and shifted up by 1 unit.
2. **Finding key points**:
- At \( x = 0 \): \( g(0) = 1 - |0| = 1 \) (Point: \( (0, 1) \))
- At \( x = 1 \): \( g(1) = 1 - |1| = 0 \) (Point: \( (1, 0) \))
- At \( x = -1 \): \( g(-1) = 1 - |-1| = 0 \) (Point: \( (-1, 0) \))
3. **Graphing**: The graph will also have a V-shape, opening downwards, with the vertex at \( (0, 1) \) and intersecting the x-axis at \( (1, 0) \) and \( (-1, 0) \).

### Step 3: Shading the Region Between the Graphs

1. **Identifying the region**: The area between \( f(x) \) and \( g(x) \) lies between the points where both functions intersect, which is at \( x = -1 \) and \( x = 1 \).
2. **Shading the area**: The region between the two graphs from \( x = -1 \) to \( x = 1 \) is the area we need to calculate.

### Step 4: Finding the Area of the Shaded Region

1. **Setting up the integral**: The area \( A \) between the curves from \( x = -1 \) to \( x = 1 \) can be found using the integral:
\[ A = \int_{-1}^{1} (g(x) - f(x)) \, dx \]
Substitute \( g(x) \) and \( f(x) \):
\[ A = \int_{-1}^{1} \left( (1 - |x|) - (|x| - 1) \right) \, dx \]
Simplifying the integrand:
\[ A = \int_{-1}^{1} (2 - 2|x|) \, dx \]

2. **Calculating the integral**: Since the function is even, we can calculate from \( 0 \) to \( 1 \) and double it:
\[ A = 2 \int_{0}^{1} (2 - 2x) \, dx \]
\[ = 2 \left[ 2x - x^2 \right]_{0}^{1} = 2 \left[ (2 \cdot 1 - 1^2) - (2 \cdot 0 - 0^2) \right] \]
\[ = 2 \left[ 2 - 1 \right] = 2 \cdot 1 = 2 \]

### Final Answer

The area of the shaded region between the two graphs is \( \boxed{2} \) square units.

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