The domain of the function ` f(x) = sqrt( ""^(x^2 + 4x )C_( 2x^2 +3))` are

Options

A. `{1,2,3}`
B. `[1,2]`
C. `[sqrt(3),2]`
D. None

Answer

To find the domain of the function \( f(x) = \sqrt{C(x^2 + 4x, 2x^2 + 3)} \), we need to ensure that the expression inside the square root is valid. This involves checking the conditions for combinations, which are defined for non-negative integers.

### Step 1: Identify the conditions for combinations
The combination \( C(n, r) \) is defined when:
1. \( n \geq 0 \)
2. \( r \geq 0 \)
3. \( n \geq r \)

Here, we have:
- \( n = x^2 + 4x \)
- \( r = 2x^2 + 3 \)

### Step 2: Set up inequalities based on the conditions
From the conditions above, we derive the following inequalities:
1. \( x^2 + 4x \geq 0 \)
2. \( 2x^2 + 3 \geq 0 \)
3. \( x^2 + 4x \geq 2x^2 + 3 \)

### Step 3: Solve the first inequality \( x^2 + 4x \geq 0 \)
Factoring gives:
\[ x(x + 4) \geq 0 \]
The roots are \( x = 0 \) and \( x = -4 \). Analyzing the sign:
- For \( x < -4 \), the product is positive.
- For \( -4 < x < 0 \), the product is negative.
- For \( x > 0 \), the product is positive.

Thus, the solution is:
\[ x \in (-\infty, -4] \cup [0, \infty) \]

### Step 4: Solve the second inequality \( 2x^2 + 3 \geq 0 \)
This inequality is always true since \( 2x^2 \) is non-negative for all \( x \) and \( 3 \) is positive. Therefore:
\[ x \in \mathbb{R} \]

### Step 5: Solve the third inequality \( x^2 + 4x \geq 2x^2 + 3 \)
Rearranging gives:
\[ -x^2 + 4x - 3 \geq 0 \]
Factoring yields:
\[ -(x - 3)(x + 1) \geq 0 \]
The roots are \( x = 3 \) and \( x = -1 \). Analyzing the sign:
- For \( x < -1 \), the product is positive.
- For \( -1 < x < 3 \), the product is negative.
- For \( x > 3 \), the product is positive.

Thus, the solution is:
\[ x \in (-\infty, -1] \cup [3, \infty) \]

### Step 6: Find the intersection of the solutions
Now we need to find the intersection of:
1. \( x \in (-\infty, -4] \cup [0, \infty) \)
2. \( x \in \mathbb{R} \)
3. \( x \in (-\infty, -1] \cup [3, \infty) \)

The intersection gives:
- From \( (-\infty, -4] \) and \( (-\infty, -1] \), we have \( (-\infty, -4] \).
- From \( [0, \infty) \) and \( [3, \infty) \), we have \( [3, \infty) \).

Thus, the overall intersection is:
\[ x \in (-\infty, -4] \cup [3, \infty) \]

### Step 7: Identify the integer values in the domain
Since \( n \) and \( r \) must be non-negative integers, we need to consider integer values:
- From \( (-\infty, -4] \): integers are \( \ldots, -5, -4 \)
- From \( [3, \infty) \): integers are \( 3, 4, 5, \ldots \)

### Final Domain
The domain of \( f(x) \) in terms of integers is:
\[ \text{Domain} = \{ \ldots, -5, -4, 3, 4, 5, \ldots \} \]