The functions r is defined as `g(x)=(x^(2))/(3-|x-4|)`. For which values of x is g(x) NOT defined?

Options

A. x=4 and x=7
B. x=3 and x=4
C. x=3 and x=7
D. x=1 and x=7

Answer

To determine the values of \( x \) for which the function \( g(x) = \frac{x^2}{3 - |x - 4|} \) is not defined, we need to identify when the denominator is equal to zero.

### Step-by-step Solution:

1. **Identify the Denominator**:
The function \( g(x) \) is defined as:
\[ g(x) = \frac{x^2}{3 - |x - 4|} \]
The function is not defined when the denominator is zero:
\[ 3 - |x - 4| = 0 \]

2. **Set the Denominator to Zero**:
Rearranging the equation gives:
\[ |x - 4| = 3 \]

3. **Solve the Absolute Value Equation**:
The absolute value equation \( |x - 4| = 3 \) can be split into two cases:
- Case 1: \( x - 4 = 3 \)
- Case 2: \( x - 4 = -3 \)

4. **Solve Case 1**:
From Case 1:
\[ x - 4 = 3 \implies x = 3 + 4 = 7 \]

5. **Solve Case 2**:
From Case 2:
\[ x - 4 = -3 \implies x = -3 + 4 = 1 \]

6. **Conclusion**:
The values of \( x \) for which \( g(x) \) is not defined are:
\[ x = 1 \quad \text{and} \quad x = 7 \]

### Final Answer:
The function \( g(x) \) is not defined for \( x = 1 \) and \( x = 7 \).