Following questions are based on the given information for the following functions f (x)
`f(x) =2bx +f(-x)," if " x lt 0`
`f(x)=a, " if "x =0`
`f(x)=b+c-2cx +f(x-1)," if "x gt 0`
If a = 15, b = 11, c = – 3, then f (7) equals:

Options

A. 239
B. 115
C. `-147`
D. None of these

Answer

To find \( f(7) \) given the piecewise function definitions and the values of \( a \), \( b \), and \( c \), we will follow these steps:

### Step 1: Identify the function definitions
The function \( f(x) \) is defined as:
- \( f(x) = 2bx + f(-x) \) for \( x < 0 \)
- \( f(x) = a \) for \( x = 0 \)
- \( f(x) = b + c - 2cx + f(x-1) \) for \( x > 0 \)

### Step 2: Substitute the values of \( a \), \( b \), and \( c \)
We are given:
- \( a = 15 \)
- \( b = 11 \)
- \( c = -3 \)

### Step 3: Calculate \( f(0) \)
Since \( f(0) = a \):
\[ f(0) = 15 \]

### Step 4: Calculate \( f(1) \)
Using the definition for \( x > 0 \):
\[ f(1) = b + c - 2c(1) + f(0) \]
Substituting the values:
\[ f(1) = 11 - 3 - 2(-3)(1) + 15 \]
Calculating:
\[ f(1) = 11 - 3 + 6 + 15 = 29 \]

### Step 5: Calculate \( f(2) \)
Using the definition for \( x > 0 \):
\[ f(2) = b + c - 2c(2) + f(1) \]
Substituting the values:
\[ f(2) = 11 - 3 - 2(-3)(2) + 29 \]
Calculating:
\[ f(2) = 11 - 3 + 12 + 29 = 49 \]

### Step 6: Calculate \( f(3) \)
Using the definition for \( x > 0 \):
\[ f(3) = b + c - 2c(3) + f(2) \]
Substituting the values:
\[ f(3) = 11 - 3 - 2(-3)(3) + 49 \]
Calculating:
\[ f(3) = 11 - 3 + 18 + 49 = 75 \]

### Step 7: Calculate \( f(4) \)
Using the definition for \( x > 0 \):
\[ f(4) = b + c - 2c(4) + f(3) \]
Substituting the values:
\[ f(4) = 11 - 3 - 2(-3)(4) + 75 \]
Calculating:
\[ f(4) = 11 - 3 + 24 + 75 = 107 \]

### Step 8: Calculate \( f(5) \)
Using the definition for \( x > 0 \):
\[ f(5) = b + c - 2c(5) + f(4) \]
Substituting the values:
\[ f(5) = 11 - 3 - 2(-3)(5) + 107 \]
Calculating:
\[ f(5) = 11 - 3 + 30 + 107 = 145 \]

### Step 9: Calculate \( f(6) \)
Using the definition for \( x > 0 \):
\[ f(6) = b + c - 2c(6) + f(5) \]
Substituting the values:
\[ f(6) = 11 - 3 - 2(-3)(6) + 145 \]
Calculating:
\[ f(6) = 11 - 3 + 36 + 145 = 189 \]

### Step 10: Calculate \( f(7) \)
Using the definition for \( x > 0 \):
\[ f(7) = b + c - 2c(7) + f(6) \]
Substituting the values:
\[ f(7) = 11 - 3 - 2(-3)(7) + 189 \]
Calculating:
\[ f(7) = 11 - 3 + 42 + 189 = 239 \]

### Final Answer
Thus, \( f(7) = 239 \).

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