Let `f : R to R` be defined by
`f(x) = ax + b AA x in R`
Where `a,b in R` and `a ne 1`
If `(fofofofof)(x) `= `32x + 93`, then value of b is ………….

Answer

To solve the problem, we need to find the value of \( b \) given that \( f(x) = ax + b \) and \( (f \circ f \circ f \circ f \circ f)(x) = 32x + 93 \).

### Step-by-Step Solution:

1. **Understanding the Function**:
We start with the function \( f(x) = ax + b \).

2. **Finding \( f(f(x)) \)**:
We substitute \( f(x) \) into itself:
\[ f(f(x)) = f(ax + b) = a(ax + b) + b = a^2x + ab + b \]
This can be rewritten as:
\[ f(f(x)) = a^2x + (ab + b) \]

3. **Finding \( f(f(f(x))) \)**:
Now, we substitute \( f(f(x)) \) into \( f \):
\[ f(f(f(x))) = f(a^2x + (ab + b)) = a(a^2x + (ab + b)) + b = a^3x + a(ab + b) + b \]
Simplifying this gives:
\[ f(f(f(x))) = a^3x + a^2b + ab + b \]

4. **Finding \( f(f(f(f(x)))) \)**:
Next, we substitute \( f(f(f(x))) \) into \( f \):
\[ f(f(f(f(x)))) = f(a^3x + (a^2b + ab + b)) = a(a^3x + (a^2b + ab + b)) + b \]
This simplifies to:
\[ f(f(f(f(x)))) = a^4x + a(a^2b + ab + b) + b \]

5. **Finding \( f(f(f(f(f(x))))) \)**:
Finally, we substitute \( f(f(f(f(x)))) \) into \( f \):
\[ f(f(f(f(f(x))))) = f(a^4x + (a(a^2b + ab + b) + b)) = a(a^4x + (a(a^2b + ab + b) + b)) + b \]
This gives:
\[ f(f(f(f(f(x))))) = a^5x + a^4b + a(a^2b + ab + b) + b \]

6. **Combining Terms**:
We can combine the terms involving \( b \):
\[ f(f(f(f(f(x))))) = a^5x + (a^4b + a^3b + a^2b + ab + b) \]
This can be factored as:
\[ f(f(f(f(f(x))))) = a^5x + b(a^4 + a^3 + a^2 + a + 1) \]

7. **Setting Equal to Given Function**:
We know that:
\[ f(f(f(f(f(x))))) = 32x + 93 \]
Therefore, we can equate coefficients:
\[ a^5 = 32 \quad \text{and} \quad b(a^4 + a^3 + a^2 + a + 1) = 93 \]

8. **Solving for \( a \)**:
From \( a^5 = 32 \), we find:
\[ a = 2 \]

9. **Substituting \( a \) to Find \( b \)**:
Now substituting \( a = 2 \) into the equation for \( b \):
\[ b(2^4 + 2^3 + 2^2 + 2 + 1) = 93 \]
Calculating the sum:
\[ 16 + 8 + 4 + 2 + 1 = 31 \]
Thus:
\[ b \cdot 31 = 93 \implies b = \frac{93}{31} = 3 \]

### Final Answer:
The value of \( b \) is \( \boxed{3} \).