Find the domain of the definition of the function
`y=log_(10) [(x-5)//(x^(2)-10x +24)] -(x+4)^(1//2)`

Options

A. `x gt 6`
B. `4 lt x`
C. Both (a) and (b)
D. None of these

Answer

To find the domain of the function \( y = \log_{10} \left( \frac{x-5}{x^2 - 10x + 24} \right) - \sqrt{x + 4} \), we need to ensure that the expression inside the logarithm is positive and that the square root is defined.

### Step 1: Determine when the logarithm is defined
The logarithm \( \log_{10}(A) \) is defined when \( A > 0 \). Here, \( A = \frac{x-5}{x^2 - 10x + 24} \).

1. **Numerator Condition**:
\[ x - 5 > 0 \implies x > 5 \]

2. **Denominator Condition**:
We need to ensure that the denominator is not zero and also that the entire fraction is positive. First, let's find when the denominator is zero:
\[ x^2 - 10x + 24 = 0 \]
We can factor this quadratic:
\[ (x - 6)(x - 4) = 0 \]
Thus, the roots are \( x = 4 \) and \( x = 6 \). The denominator is zero at these points, so we need to exclude them from the domain.

### Step 2: Analyze the sign of the fraction
Next, we analyze the sign of the fraction \( \frac{x-5}{(x-6)(x-4)} \).

- **Intervals to test**: We will test the intervals determined by the critical points \( 4 \) and \( 6 \):
- \( (-\infty, 4) \)
- \( (4, 5) \)
- \( (5, 6) \)
- \( (6, \infty) \)

1. **For \( x < 4 \)** (e.g., \( x = 0 \)):
\[ \frac{0-5}{(0-6)(0-4)} = \frac{-5}{24} < 0 \]

2. **For \( 4 < x < 5 \)** (e.g., \( x = 4.5 \)):
\[ \frac{4.5-5}{(4.5-6)(4.5-4)} = \frac{-0.5}{(-1.5)(0.5)} > 0 \]

3. **For \( 5 < x < 6 \)** (e.g., \( x = 5.5 \)):
\[ \frac{5.5-5}{(5.5-6)(5.5-4)} = \frac{0.5}{(-0.5)(1.5)} < 0 \]

4. **For \( x > 6 \)** (e.g., \( x = 7 \)):
\[ \frac{7-5}{(7-6)(7-4)} = \frac{2}{1 \cdot 3} > 0 \]

### Step 3: Combine conditions
From the analysis:
- The fraction is positive in the intervals \( (4, 5) \) and \( (6, \infty) \).
- However, since \( x > 5 \) is also required, we can only take the interval \( (6, \infty) \).

### Step 4: Ensure the square root is defined
The square root \( \sqrt{x + 4} \) is defined for:
\[ x + 4 \geq 0 \implies x \geq -4 \]
This condition is satisfied for all \( x > 6 \).

### Conclusion
Thus, the domain of the function is:
\[ \text{Domain: } (6, \infty) \]