Define the following functions:
(i) `a "@ "b =(a+b)/(2)`
(ii) `a# b =a^(2) -b^(2)`
(iii) `(a!b)=(a-b)/(2)`
Find the value of {[(3@4)!(3#2)]@[(4!3)@(2#3)]}

Options

A. `-0.75`
B. `-1`
C. `-1.5`
D. `-2.25`

Answer

To solve the problem step by step, we will evaluate the functions defined in the question and then substitute them into the expression provided.

### Step 1: Define the Functions
We have three functions defined as follows:
1. \( a \, @ \, b = \frac{a + b}{2} \)
2. \( a \, \# \, b = a^2 - b^2 \)
3. \( a \, ! \, b = \frac{a - b}{2} \)

### Step 2: Evaluate the Inner Expressions
We need to evaluate the expression:
\[ \{[(3 \, @ \, 4) \, ! \, (3 \, \# \, 2)] \, @ \, [(4 \, ! \, 3) \, @ \, (2 \, \# \, 3)]\} \]

#### Step 2.1: Calculate \( 3 \, @ \, 4 \)
\[ 3 \, @ \, 4 = \frac{3 + 4}{2} = \frac{7}{2} \]

#### Step 2.2: Calculate \( 3 \, \# \, 2 \)
\[ 3 \, \# \, 2 = 3^2 - 2^2 = 9 - 4 = 5 \]

#### Step 2.3: Calculate \( (3 \, @ \, 4) \, ! \, (3 \, \# \, 2) \)
\[ (3 \, @ \, 4) \, ! \, (3 \, \# \, 2) = \left(\frac{7}{2}\right) ! 5 = \frac{\frac{7}{2} - 5}{2} = \frac{\frac{7}{2} - \frac{10}{2}}{2} = \frac{-\frac{3}{2}}{2} = -\frac{3}{4} \]

#### Step 2.4: Calculate \( 4 \, ! \, 3 \)
\[ 4 \, ! \, 3 = \frac{4 - 3}{2} = \frac{1}{2} \]

#### Step 2.5: Calculate \( 2 \, \# \, 3 \)
\[ 2 \, \# \, 3 = 2^2 - 3^2 = 4 - 9 = -5 \]

#### Step 2.6: Calculate \( (4 \, ! \, 3) \, @ \, (2 \, \# \, 3) \)
\[ (4 \, ! \, 3) \, @ \, (2 \, \# \, 3) = \left(\frac{1}{2}\right) @ (-5) = \frac{\frac{1}{2} + (-5)}{2} = \frac{\frac{1}{2} - \frac{10}{2}}{2} = \frac{-\frac{9}{2}}{2} = -\frac{9}{4} \]

### Step 3: Final Calculation
Now we need to evaluate:
\[ \left(-\frac{3}{4}\right) \, @ \left(-\frac{9}{4}\right) \]
\[ -\frac{3}{4} \, @ \, -\frac{9}{4} = \frac{-\frac{3}{4} + (-\frac{9}{4})}{2} = \frac{-\frac{12}{4}}{2} = \frac{-3}{2} = -1.5 \]

### Final Answer
The value of the expression is:
\[ \boxed{-1.5} \]