The domain of the function `f(x)=log_(2)[1-log_(12)(x^(2)-5x+16)]` is

Options

A. `(1, 4)`
B. `(-oo, 4]`
C. `[1, oo)`
D. `[1, 4]`

Answer

To find the domain of the function \( f(x) = \log_2[1 - \log_{12}(x^2 - 5x + 16)] \), we need to ensure that the expression inside the logarithm is defined and valid.

### Step 1: Identify the conditions for the logarithm
The logarithmic function is defined only for positive arguments. Therefore, we need:
1. \( 1 - \log_{12}(x^2 - 5x + 16) > 0 \)
2. \( x^2 - 5x + 16 > 0 \)

### Step 2: Solve the first inequality
From the first inequality:
\[ 1 - \log_{12}(x^2 - 5x + 16) > 0 \]
This simplifies to:
\[ \log_{12}(x^2 - 5x + 16) < 1 \]
Taking the antilogarithm gives:
\[ x^2 - 5x + 16 < 12^1 \]
which simplifies to:
\[ x^2 - 5x + 16 < 12 \]
Rearranging this gives:
\[ x^2 - 5x + 4 < 0 \]

### Step 3: Factor the quadratic
Now we need to factor the quadratic:
\[ x^2 - 5x + 4 = (x - 4)(x - 1) \]
Thus, we have:
\[ (x - 4)(x - 1) < 0 \]

### Step 4: Determine the intervals
To find the intervals where this product is negative, we can test the intervals determined by the roots \( x = 1 \) and \( x = 4 \):
- For \( x < 1 \): Choose \( x = 0 \) → \( (0 - 4)(0 - 1) = 4 > 0 \)
- For \( 1 < x < 4 \): Choose \( x = 2 \) → \( (2 - 4)(2 - 1) = -2 < 0 \)
- For \( x > 4 \): Choose \( x = 5 \) → \( (5 - 4)(5 - 1) = 4 > 0 \)

Thus, the solution to \( (x - 4)(x - 1) < 0 \) is:
\[ 1 < x < 4 \]

### Step 5: Check the second inequality
Now we check the second condition \( x^2 - 5x + 16 > 0 \). The quadratic \( x^2 - 5x + 16 \) has a discriminant of:
\[ D = (-5)^2 - 4 \cdot 1 \cdot 16 = 25 - 64 = -39 \]
Since the discriminant is negative, the quadratic is always positive for all \( x \).

### Conclusion
The domain of the function \( f(x) \) is:
\[ (1, 4) \]