The domain of the function `f(x)=1/(sqrt(4x-|x^2-10 x+9|))` is (a)`(7-sqrt(40),7+sqrt(40))`(b) `(0,7+sqrt(40))`(c)`(7-sqrt(40),oo)` (d) none of these

Options

A. `(7-sqrt(40), 7+sqrt(40))`
B. `(0,7+sqrt(40))`
C. `(7-sqrt(40),oo)`
D. none of these

Explanation

To find the domain of the function \( f(x) = \frac{1}{\sqrt{4x - |x^2 - 10x + 9|}} \), we need to ensure that the expression inside the square root is positive. This means we need to solve the inequality:

\[ 4x - |x^2 - 10x + 9| > 0 \]

### Step 1: Analyze the expression inside the absolute value

First, we need to simplify the expression \( |x^2 - 10x + 9| \). We can factor the quadratic:

\[ x^2 - 10x + 9 = (x - 1)(x - 9) \]

### Step 2: Determine the critical points

The critical points where the expression changes sign are found by setting \( x^2 - 10x + 9 = 0 \):

\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]

### Step 3: Test intervals

We will test the intervals defined by the critical points \( x = 1 \) and \( x = 9 \):

1. **Interval \( (-\infty, 1) \)**:
- Choose \( x = 0 \):
\[ |0^2 - 10(0) + 9| = |9| = 9 \]
\[ 4(0) - 9 = -9 \quad \text{(not valid)} \]

2. **Interval \( (1, 9) \)**:
- Choose \( x = 5 \):
\[ |5^2 - 10(5) + 9| = |25 - 50 + 9| = |-16| = 16 \]
\[ 4(5) - 16 = 20 - 16 = 4 \quad \text{(valid)} \]

3. **Interval \( (9, \infty) \)**:
- Choose \( x = 10 \):
\[ |10^2 - 10(10) + 9| = |100 - 100 + 9| = |9| = 9 \]
\[ 4(10) - 9 = 40 - 9 = 31 \quad \text{(valid)} \]

### Step 4: Combine results

From the tests, we find that the expression \( 4x - |x^2 - 10x + 9| > 0 \) is valid in the intervals \( (1, 9) \) and \( (9, \infty) \).

### Step 5: Exclude points where the expression is zero

We also need to check where the expression inside the square root equals zero:

1. Set \( 4x - |x^2 - 10x + 9| = 0 \):
- For \( x \in (1, 9) \):
\[ 4x = |x^2 - 10x + 9| \Rightarrow 4x = 16 \quad \Rightarrow x = 4 \]
- For \( x \in (9, \infty) \):
\[ 4x = 9 \quad \Rightarrow x = \frac{9}{4} \quad \text{(not in this interval)} \]

Thus, \( x = 4 \) must be excluded from the domain.

### Final Domain

The domain of \( f(x) \) is:

\[ (1, 4) \cup (4, 9) \cup (9, \infty) \]

### Conclusion

Therefore, the correct answer is:

**(d) none of these**