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

Options

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

Answer

To find the range of the function \( f(x) = \sqrt{ \binom{x^2 + 4x}{2x^2 + 3} } \), we will follow these steps:

### Step 1: Identify the conditions for the binomial coefficient
The binomial coefficient \( \binom{n}{r} \) is defined when:
1. \( n \) must be a non-negative integer.
2. \( r \) must be a non-negative integer.
3. \( n \) must be greater than or equal to \( r \).

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

### Step 2: Set up inequalities based on the conditions
1. **Condition 1**: \( n \geq 0 \)
\[ x^2 + 4x \geq 0 \]
Factoring gives:
\[ x(x + 4) \geq 0 \]
The roots are \( x = 0 \) and \( x = -4 \). Testing intervals, we find:
- Satisfied for \( x \in (-\infty, -4] \cup [0, \infty) \).

2. **Condition 2**: \( r \geq 0 \)
\[ 2x^2 + 3 \geq 0 \]
This is always satisfied for all \( x \in \mathbb{R} \) since \( 2x^2 \) is always non-negative.

3. **Condition 3**: \( n \geq r \)
\[ x^2 + 4x \geq 2x^2 + 3 \]
Rearranging gives:
\[ -x^2 + 4x - 3 \geq 0 \quad \Rightarrow \quad x^2 - 4x + 3 \leq 0 \]
Factoring gives:
\[ (x - 1)(x - 3) \leq 0 \]
The roots are \( x = 1 \) and \( x = 3 \). Testing intervals, we find:
- Satisfied for \( x \in [1, 3] \).

### Step 3: Determine the intersection of the intervals
From the conditions:
- From Condition 1: \( x \in (-\infty, -4] \cup [0, \infty) \)
- From Condition 3: \( x \in [1, 3] \)

The intersection of these intervals is:
\[ x \in [1, 3] \]

### Step 4: Calculate the values of \( f(x) \) at the endpoints and within the interval
1. **At \( x = 1 \)**:
\[ f(1) = \sqrt{ \binom{1^2 + 4(1)}{2(1^2) + 3} } = \sqrt{ \binom{5}{5} } = \sqrt{1} = 1 \]

2. **At \( x = 2 \)**:
\[ f(2) = \sqrt{ \binom{2^2 + 4(2)}{2(2^2) + 3} } = \sqrt{ \binom{12}{11} } = \sqrt{12} = 2\sqrt{3} \]

3. **At \( x = 3 \)**:
\[ f(3) = \sqrt{ \binom{3^2 + 4(3)}{2(3^2) + 3} } = \sqrt{ \binom{21}{21} } = \sqrt{1} = 1 \]

### Step 5: Compile the range
The values obtained are:
- \( f(1) = 1 \)
- \( f(2) = 2\sqrt{3} \)
- \( f(3) = 1 \)

Thus, the range of \( f(x) \) is:
\[ \text{Range} = \{ 1, 2\sqrt{3} \} \]

### Final Answer
The range of the function \( f(x) \) is \( \{ 1, 2\sqrt{3} \} \).

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