`a#b=(-1)^(ab)(a^(b)+b^(a))`
`f(x)=x^(2)-2x if x ge 0`
`=0if x lt0`
`g(x)=2x, if x ge0`
`=1, if x lt0`
Find the value of `f(g(2#3))+g(f(1#2)):`

Options

A. 9401
B. 1049
C. 1094
D. none of these

Answer

To solve the problem, we need to evaluate the expression \( f(g(2 \# 3)) + g(f(1 \# 2)) \). We will break this down step by step.

### Step 1: Calculate \( 2 \# 3 \)

The operation \( a \# b \) is defined as:

\[ a \# b = (-1)^{ab} (a^b + b^a) \]

Substituting \( a = 2 \) and \( b = 3 \):

\[ 2 \# 3 = (-1)^{2 \cdot 3} (2^3 + 3^2) \]

Calculating the exponent:

\[ (-1)^{6} = 1 \quad (\text{since 6 is even}) \]

Now calculate \( 2^3 \) and \( 3^2 \):

\[ 2^3 = 8 \quad \text{and} \quad 3^2 = 9 \]

Adding these results together:

\[ 2 \# 3 = 1 \cdot (8 + 9) = 1 \cdot 17 = 17 \]

### Step 2: Calculate \( g(2 \# 3) = g(17) \)

The function \( g(x) \) is defined as:

\[ g(x) = \begin{cases} 2x & \text{if } x \geq 0 \\ 1 & \text{if } x < 0 \end{cases} \]

Since \( 17 \geq 0 \):

\[ g(17) = 2 \cdot 17 = 34 \]

### Step 3: Calculate \( f(g(2 \# 3)) = f(34) \)

The function \( f(x) \) is defined as:

\[ f(x) = \begin{cases} x^2 - 2x & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases} \]

Since \( 34 \geq 0 \):

\[ f(34) = 34^2 - 2 \cdot 34 \]

Calculating \( 34^2 \):

\[ 34^2 = 1156 \]

Calculating \( 2 \cdot 34 \):

\[ 2 \cdot 34 = 68 \]

Now, substituting back:

\[ f(34) = 1156 - 68 = 1088 \]

### Step 4: Calculate \( 1 \# 2 \)

Using the same operation definition:

\[ 1 \# 2 = (-1)^{1 \cdot 2} (1^2 + 2^1) \]

Calculating the exponent:

\[ (-1)^{2} = 1 \quad (\text{since 2 is even}) \]

Calculating \( 1^2 \) and \( 2^1 \):

\[ 1^2 = 1 \quad \text{and} \quad 2^1 = 2 \]

Adding these results together:

\[ 1 \# 2 = 1 \cdot (1 + 2) = 1 \cdot 3 = 3 \]

### Step 5: Calculate \( f(1 \# 2) = f(3) \)

Since \( 3 \geq 0 \):

\[ f(3) = 3^2 - 2 \cdot 3 \]

Calculating \( 3^2 \):

\[ 3^2 = 9 \]

Calculating \( 2 \cdot 3 \):

\[ 2 \cdot 3 = 6 \]

Now, substituting back:

\[ f(3) = 9 - 6 = 3 \]

### Step 6: Calculate \( g(f(1 \# 2)) = g(3) \)

Since \( 3 \geq 0 \):

\[ g(3) = 2 \cdot 3 = 6 \]

### Step 7: Combine the results

Now we need to find:

\[ f(g(2 \# 3)) + g(f(1 \# 2)) = f(34) + g(3) = 1088 + 6 = 1094 \]

### Final Answer

The final answer is:

\[ \boxed{1094} \]