If `f(x) = (1)/(sqrt( (x+1) (e^(x) - 1)(x-4)(x +5)(x -6)))`, then the domain of `f(x)` is

Options

A. `(infty, -5), cup (-1,4), cup (6, infty)`
B. `(- infty, - 5) cup (- 1, 0) cup (0,4) cup (6, infty)`
C. `( - 5, -1) cup ( 0, 4) cup (6 , infty)`
D. None of these

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{(x+1)(e^x - 1)(x-4)(x+5)(x-6)}} \), we need to ensure that the expression inside the square root is positive and not equal to zero, since it is in the denominator.

### Step-by-step Solution:

1. **Identify the expression inside the square root:**
\[ (x+1)(e^x - 1)(x-4)(x+5)(x-6) \]

2. **Determine when each factor is zero:**
- \( x + 1 = 0 \) gives \( x = -1 \)
- \( e^x - 1 = 0 \) gives \( x = 0 \)
- \( x - 4 = 0 \) gives \( x = 4 \)
- \( x + 5 = 0 \) gives \( x = -5 \)
- \( x - 6 = 0 \) gives \( x = 6 \)

The critical points are \( x = -5, -1, 0, 4, 6 \).

3. **Analyze the intervals defined by these critical points:**
The critical points divide the number line into the following intervals:
- \( (-\infty, -5) \)
- \( (-5, -1) \)
- \( (-1, 0) \)
- \( (0, 4) \)
- \( (4, 6) \)
- \( (6, \infty) \)

4. **Test each interval to determine where the product is positive:**
- For \( x < -5 \) (e.g., \( x = -6 \)):
\[ (-)(-)(-)(-)(-) = - \quad \text{(Negative)} \]
- For \( -5 < x < -1 \) (e.g., \( x = -3 \)):
\[ (+)(-)(-)(+)(-) = + \quad \text{(Positive)} \]
- For \( -1 < x < 0 \) (e.g., \( x = -0.5 \)):
\[ (+)(-)(-)(+)(-) = + \quad \text{(Positive)} \]
- For \( 0 < x < 4 \) (e.g., \( x = 1 \)):
\[ (+)(+)(-)(+)(-) = - \quad \text{(Negative)} \]
- For \( 4 < x < 6 \) (e.g., \( x = 5 \)):
\[ (+)(+)(+)(+)(-) = - \quad \text{(Negative)} \]
- For \( x > 6 \) (e.g., \( x = 7 \)):
\[ (+)(+)(+)(+)(+) = + \quad \text{(Positive)} \]

5. **Combine the intervals where the product is positive:**
The intervals where the expression is positive are:
- From \( (-5, -1) \)
- From \( (-1, 0) \)
- From \( (6, \infty) \)

6. **Write the domain of \( f(x) \):**
Therefore, the domain of \( f(x) \) is:
\[ (-5, -1) \cup (-1, 0) \cup (6, \infty) \]

### Final Answer:
The domain of \( f(x) \) is \( (-5, -1) \cup (-1, 0) \cup (6, \infty) \).