The function f(x) is defined as follows:
`{:(,f(x) = x^(2) -1 , if x le 3),(,f(x) = 2x + 2 , if 3 lt x le 9),(,f(x) = 4x - 8, if x gt 9):}`
What is the value of k if `f(f(f(3))) = (k+1)^(2)` where k is positive integer ?

Answer

To solve the problem, we need to evaluate \( f(f(f(3))) \) using the piecewise function defined for \( f(x) \). Let's break this down step by step.

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

Since \( 3 \leq 3 \), we use the first piece of the function:
\[ f(x) = x^2 - 1 \]
Substituting \( x = 3 \):
\[ f(3) = 3^2 - 1 = 9 - 1 = 8 \]

### Step 2: Calculate \( f(f(3)) = f(8) \)

Next, we need to find \( f(8) \). Since \( 3 < 8 \leq 9 \), we use the second piece of the function:
\[ f(x) = 2x + 2 \]
Substituting \( x = 8 \):
\[ f(8) = 2 \cdot 8 + 2 = 16 + 2 = 18 \]

### Step 3: Calculate \( f(f(f(3))) = f(18) \)

Now we need to find \( f(18) \). Since \( 18 > 9 \), we use the third piece of the function:
\[ f(x) = 4x - 8 \]
Substituting \( x = 18 \):
\[ f(18) = 4 \cdot 18 - 8 = 72 - 8 = 64 \]

### Step 4: Set up the equation \( f(f(f(3))) = (k + 1)^2 \)

We have found that:
\[ f(f(f(3))) = 64 \]
We need to find \( k \) such that:
\[ 64 = (k + 1)^2 \]

### Step 5: Solve for \( k \)

Taking the square root of both sides:
\[ k + 1 = 8 \quad \text{or} \quad k + 1 = -8 \]
Since \( k \) is a positive integer, we only consider:
\[ k + 1 = 8 \implies k = 8 - 1 = 7 \]

### Final Answer

Thus, the value of \( k \) is:
\[ \boxed{7} \]