Let `f` and `g` be real valued functions defined as
`f(x)={{:(7x^(2)+x-8",",xle 1),(4x+5",",1lt x le 7),(8x+3",",x gt7):} " " g(x)={{:(|x|",",xlt -3),(0",",-3le x lt 2),(x^(2)+4",",xge 2):}`
The value of `gof (0) + fog (-3)` is

Options

A. a) -8
B. b) 0
C. c) 8
D. d) 16

Explanation

To solve the problem, we need to find the value of \( g(f(0)) + f(g(-3)) \).

### Step 1: Calculate \( f(0) \)
The function \( f(x) \) is defined as:
- \( f(x) = 7x^2 + x - 8 \) for \( x \leq 1 \)

Since \( 0 \leq 1 \), we will use this part of the function:
\[ f(0) = 7(0)^2 + 0 - 8 = 0 - 8 = -8 \]

### Step 2: Calculate \( g(f(0)) = g(-8) \)
Now we need to find \( g(-8) \). The function \( g(x) \) is defined as:
- \( g(x) = |x| \) for \( x < -3 \)

Since \( -8 < -3 \), we will use this part of the function:
\[ g(-8) = |-8| = 8 \]

### Step 3: Calculate \( g(-3) \)
Next, we need to find \( g(-3) \):
- \( g(x) = 0 \) for \( -3 \leq x < 2 \)

Since \( -3 \) is included in this range:
\[ g(-3) = 0 \]

### Step 4: Calculate \( f(g(-3)) = f(0) \)
Now we need to find \( f(g(-3)) = f(0) \):
\[ f(0) = -8 \quad \text{(as calculated in Step 1)} \]

### Step 5: Combine the results
Now we can combine the results:
\[ g(f(0)) + f(g(-3)) = g(-8) + f(0) = 8 + (-8) = 0 \]

### Final Answer
Thus, the value of \( g(f(0)) + f(g(-3)) \) is \( 0 \).