Define the following functions:
(i) `a "@"b =(a+b)/(2)`
(ii) `a# b =a^(2) -b^(2)`
(iii) `(a!b)=(a-b)/(2)`
Which of the following expressions has a value of 4 for a = 5 and b = 3?

Options

A. `((a!b))/((a#b))`
B. `(a!b) (a"@"b)`
C. `((a#b))/((a!b)(a"@"b))`
D. Both (b) and (c)

Answer

To solve the problem, we need to evaluate the three defined functions for \( a = 5 \) and \( b = 3 \) and check which expression yields a value of 4.

### Step 1: Evaluate the first function \( a \, @ \, b \)
The function is defined as:
\[ a \, @ \, b = \frac{a + b}{2} \]
Substituting \( a = 5 \) and \( b = 3 \):
\[ 5 \, @ \, 3 = \frac{5 + 3}{2} = \frac{8}{2} = 4 \]

### Step 2: Evaluate the second function \( a \# b \)
The function is defined as:
\[ a \# b = a^2 - b^2 \]
Substituting \( a = 5 \) and \( b = 3 \):
\[ 5 \# 3 = 5^2 - 3^2 = 25 - 9 = 16 \]

### Step 3: Evaluate the third function \( a ! b \)
The function is defined as:
\[ a ! b = \frac{a - b}{2} \]
Substituting \( a = 5 \) and \( b = 3 \):
\[ 5 ! 3 = \frac{5 - 3}{2} = \frac{2}{2} = 1 \]

### Summary of Results
- \( 5 \, @ \, 3 = 4 \)
- \( 5 \# 3 = 16 \)
- \( 5 ! 3 = 1 \)

### Conclusion
The expression that has a value of 4 for \( a = 5 \) and \( b = 3 \) is \( a \, @ \, b \).

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