Let `f(x)={{:(x^(2)-4x+3",", x lt 3),(x-4",", x ge 3):}`
and `g(x)={{:(x-3",", x lt 4),(x^(2)+2x+2",", x ge 4):}`, which one of the following is/are true?

Options

A. a) `(f+g)(3.5)=0`
B. b) `f(g(3))=3`
C. c) `f(g(2))=1`
D. d) `(f-g)(4)=0`

Answer

To solve the problem, we will evaluate each of the options given for the functions \( f(x) \) and \( g(x) \).

### Given Functions:
1. \( f(x) = \begin{cases} x^2 - 4x + 3 & \text{if } x < 3 \\ x - 4 & \text{if } x \geq 3 \end{cases} \)

2. \( g(x) = \begin{cases} x - 3 & \text{if } x < 4 \\ x^2 + 2x + 2 & \text{if } x \geq 4 \end{cases} \)

### Option A: \( f + g(3.5) = 0 \)
1. **Calculate \( f(3.5) \)**:
- Since \( 3.5 \geq 3 \), we use \( f(x) = x - 4 \).
- \( f(3.5) = 3.5 - 4 = -0.5 \).

2. **Calculate \( g(3.5) \)**:
- Since \( 3.5 < 4 \), we use \( g(x) = x - 3 \).
- \( g(3.5) = 3.5 - 3 = 0.5 \).

3. **Calculate \( f(3.5) + g(3.5) \)**:
- \( f(3.5) + g(3.5) = -0.5 + 0.5 = 0 \).

**Conclusion**: Option A is **true**.

### Option B: \( f(g(3)) = 3 \)
1. **Calculate \( g(3) \)**:
- Since \( 3 < 4 \), we use \( g(x) = x - 3 \).
- \( g(3) = 3 - 3 = 0 \).

2. **Calculate \( f(g(3)) = f(0) \)**:
- Since \( 0 < 3 \), we use \( f(x) = x^2 - 4x + 3 \).
- \( f(0) = 0^2 - 4(0) + 3 = 3 \).

**Conclusion**: Option B is **true**.

### Option C: \( f(g(2)) = 1 \)
1. **Calculate \( g(2) \)**:
- Since \( 2 < 4 \), we use \( g(x) = x - 3 \).
- \( g(2) = 2 - 3 = -1 \).

2. **Calculate \( f(g(2)) = f(-1) \)**:
- Since \( -1 < 3 \), we use \( f(x) = x^2 - 4x + 3 \).
- \( f(-1) = (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = 8 \).

**Conclusion**: Option C is **false**.

### Option D: \( f - g(4) = 0 \)
1. **Calculate \( f(4) \)**:
- Since \( 4 \geq 3 \), we use \( f(x) = x - 4 \).
- \( f(4) = 4 - 4 = 0 \).

2. **Calculate \( g(4) \)**:
- Since \( 4 \geq 4 \), we use \( g(x) = x^2 + 2x + 2 \).
- \( g(4) = 4^2 + 2(4) + 2 = 16 + 8 + 2 = 26 \).

3. **Calculate \( f(4) - g(4) \)**:
- \( f(4) - g(4) = 0 - 26 = -26 \).

**Conclusion**: Option D is **false**.

### Final Results:
- **True Options**: A and B
- **False Options**: C and D

### Summary of Results:
- Option A: True
- Option B: True
- Option C: False
- Option D: False