`f (x) (3)/((x -7) ^(2) + 6 (x -7) +9)`
For which value of x is the function f (x) underfined ?

Explanation

To find the value of \( x \) for which the function \( f(x) = \frac{3}{(x - 7)^2 + 6(x - 7) + 9} \) is undefined, we need to determine when the denominator equals zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is given by:
\[ (x - 7)^2 + 6(x - 7) + 9 \]

2. **Set the Denominator Equal to Zero**:
To find when the function is undefined, we set the denominator equal to zero:
\[ (x - 7)^2 + 6(x - 7) + 9 = 0 \]

3. **Substitute \( y = x - 7 \)**:
Let \( y = x - 7 \). Then the equation becomes:
\[ y^2 + 6y + 9 = 0 \]

4. **Factor the Quadratic**:
The quadratic can be factored as:
\[ (y + 3)^2 = 0 \]

5. **Solve for \( y \)**:
Setting the factored form equal to zero gives:
\[ y + 3 = 0 \implies y = -3 \]

6. **Substitute Back to Find \( x \)**:
Recall that \( y = x - 7 \). Therefore:
\[ x - 7 = -3 \implies x = 4 \]

### Conclusion:
The function \( f(x) \) is undefined at \( x = 4 \).