To solve the problem, we need to find the integral \( \int_2^3 (g(x) - f(x)) \, dx \) where \( f(x) = |x - 2| \) and \( g(x) = f(f(x)) \).
### Step 1: Determine \( f(x) \)
The function \( f(x) = |x - 2| \) can be expressed piecewise based on the value of \( x \):
- For \( x < 2 \): \( f(x) = 2 - x \)
- For \( x \geq 2 \): \( f(x) = x - 2 \)
### Step 2: Find \( f(x) \) for the interval [2, 3]
Since we are interested in the interval from 2 to 3:
- At \( x = 2 \): \( f(2) = |2 - 2| = 0 \)
- At \( x = 3 \): \( f(3) = |3 - 2| = 1 \)
Thus, for \( x \) in [2, 3], \( f(x) = x - 2 \).
### Step 3: Determine \( g(x) = f(f(x)) \)
Next, we need to find \( g(x) \):
1. **Calculate \( f(x) \)**:
- For \( x \in [2, 3] \): \( f(x) = x - 2 \)
2. **Calculate \( f(f(x)) \)**:
- For \( x \in [2, 3] \): \( f(x) = x - 2 \) which ranges from \( 0 \) to \( 1 \) as \( x \) goes from \( 2 \) to \( 3 \).
- Since \( f(x) \) is in the range [0, 1], we use the first case of \( f(x) \):
- \( f(f(x)) = f(x - 2) = 2 - (x - 2) = 4 - x \)
Thus, for \( x \in [2, 3] \), \( g(x) = 4 - x \).
### Step 4: Calculate \( g(x) - f(x) \)
Now we can find \( g(x) - f(x) \):
\[
g(x) - f(x) = (4 - x) - (x - 2) = 4 - x - x + 2 = 6 - 2x
\]
### Step 5: Set up the integral
We need to evaluate the integral:
\[
\int_2^3 (g(x) - f(x)) \, dx = \int_2^3 (6 - 2x) \, dx
\]
### Step 6: Evaluate the integral
Calculating the integral:
\[
\int (6 - 2x) \, dx = 6x - x^2 + C
\]
Now we evaluate from 2 to 3:
\[
\left[ 6x - x^2 \right]_2^3 = \left( 6(3) - (3)^2 \right) - \left( 6(2) - (2)^2 \right)
\]
Calculating the values:
1. At \( x = 3 \):
\[
6(3) - (3)^2 = 18 - 9 = 9
\]
2. At \( x = 2 \):
\[
6(2) - (2)^2 = 12 - 4 = 8
\]
Thus, the integral evaluates to:
\[
9 - 8 = 1
\]
### Final Answer
The value of the integral \( \int_2^3 (g(x) - f(x)) \, dx \) is \( 1 \).
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