The domain of `f(x)=log(5x-6-x^(2))` is :

Options

A. `(5, 6)`
B. `(2, 3)`
C. `[2, oo)`
D. none of these

Answer

To find the domain of the function \( f(x) = \log(5x - 6 - x^2) \), we need to ensure that the argument of the logarithm is positive. This means we need to solve the inequality:

\[ 5x - 6 - x^2 > 0 \]

### Step 1: Rearranging the Inequality
Rearranging the inequality gives us:

\[ -x^2 + 5x - 6 > 0 \]

We can multiply through by -1 (remembering to flip the inequality sign):

\[ x^2 - 5x + 6 < 0 \]

### Step 2: Factoring the Quadratic
Next, we need to factor the quadratic expression \( x^2 - 5x + 6 \). We look for two numbers that multiply to 6 and add to -5. The factors are -2 and -3:

\[ (x - 2)(x - 3) < 0 \]

### Step 3: Finding Critical Points
The critical points from the factors are \( x = 2 \) and \( x = 3 \). These points will divide the number line into intervals that we can test:

1. \( (-\infty, 2) \)
2. \( (2, 3) \)
3. \( (3, \infty) \)

### Step 4: Testing Intervals
We will test each interval to see where the product \( (x - 2)(x - 3) \) is negative.

- **Interval 1: \( (-\infty, 2) \)**
- Choose \( x = 0 \):
\[ (0 - 2)(0 - 3) = (-2)(-3) = 6 \quad (\text{positive}) \]

- **Interval 2: \( (2, 3) \)**
- Choose \( x = 2.5 \):
\[ (2.5 - 2)(2.5 - 3) = (0.5)(-0.5) = -0.25 \quad (\text{negative}) \]

- **Interval 3: \( (3, \infty) \)**
- Choose \( x = 4 \):
\[ (4 - 2)(4 - 3) = (2)(1) = 2 \quad (\text{positive}) \]

### Step 5: Conclusion
The inequality \( (x - 2)(x - 3) < 0 \) is satisfied in the interval \( (2, 3) \). Therefore, the domain of the function \( f(x) \) is:

\[ \boxed{(2, 3)} \]