`f(x)=(1)/((x-3)^(2)-6(x-3)+9)`
For what value of x is the function f defined above un defined?

Options

A. `-6`
B. `-3`
C. `3`
D. `6`

Answer

To determine for what value of \( x \) the function \( f(x) = \frac{1}{(x-3)^2 - 6(x-3) + 9} \) is undefined, we need to find when the denominator equals zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The function is undefined when the denominator is zero. Thus, we need to solve:
\[ (x-3)^2 - 6(x-3) + 9 = 0 \]

2. **Substitute for Simplicity**:
Let \( y = x - 3 \). Then, we can rewrite the equation as:
\[ y^2 - 6y + 9 = 0 \]

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

4. **Solve for \( y \)**:
Setting the factored equation to zero gives:
\[ y - 3 = 0 \implies y = 3 \]

5. **Back Substitute for \( x \)**:
Recall that \( y = x - 3 \). Therefore:
\[ x - 3 = 3 \implies x = 6 \]

6. **Conclusion**:
The function \( f(x) \) is undefined at:
\[ x = 6 \]

### Final Answer:
The function \( f(x) \) is undefined for \( x = 6 \).

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