State the domain of these functions :
`g: x to (x-4)/((x-3)(x+6))`

Answer

To find the domain of the function \( g(x) = \frac{x - 4}{(x - 3)(x + 6)} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined when the denominator is equal to zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( (x - 3)(x + 6) \).

2. **Set the Denominator Equal to Zero**:
To find the values that make the function undefined, we set the denominator equal to zero:
\[ (x - 3)(x + 6) = 0 \]

3. **Solve for \( x \)**:
We solve the equation by setting each factor equal to zero:
- \( x - 3 = 0 \) gives \( x = 3 \)
- \( x + 6 = 0 \) gives \( x = -6 \)

4. **Determine the Domain**:
The function \( g(x) \) is defined for all real numbers except where the denominator is zero. Therefore, the domain of \( g(x) \) is:
\[ \text{Domain} = \mathbb{R} \setminus \{3, -6\} \]
This means all real numbers except \( 3 \) and \( -6 \).

### Final Answer:
The domain of the function \( g(x) = \frac{x - 4}{(x - 3)(x + 6)} \) is:
\[ \text{Domain} = \{ x \in \mathbb{R} : x \neq 3 \text{ and } x \neq -6 \} \]