Range (परिसर) of `f(x) = (3)/(2-x^(2))` is

Options

A. `y ∈ (-oo,0)U[3/2,oo)`
B. `y ∈ (-oo,0]U[3/2,oo)`
C. `y ∈ (-oo,0)U(3/2,oo)`
D. None

Explanation

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

### Step 1: Identify the function and its behavior
The function \( f(x) = \frac{3}{2 - x^2} \) is a rational function. The denominator cannot be zero, so we first need to determine the values of \( x \) that make the denominator zero.

### Step 2: Find the values that make the denominator zero
Set the denominator equal to zero:
\[ 2 - x^2 = 0 \]
Solving for \( x \):
\[ x^2 = 2 \implies x = \pm \sqrt{2} \]
Thus, the function is undefined at \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

### Step 3: Determine the domain of the function
The domain of \( f(x) \) is all real numbers except \( x = \sqrt{2} \) and \( x = -\sqrt{2} \):
\[ \text{Domain} = \mathbb{R} \setminus \{ \sqrt{2}, -\sqrt{2} \} \]

### Step 4: Analyze the behavior of the function
To find the range, we analyze the behavior of \( f(x) \) as \( x \) approaches the points where the function is undefined and as \( x \) approaches infinity.

1. As \( x \to \sqrt{2} \) from the left, \( f(x) \to +\infty \).
2. As \( x \to \sqrt{2} \) from the right, \( f(x) \to -\infty \).
3. As \( x \to -\sqrt{2} \) from the left, \( f(x) \to -\infty \).
4. As \( x \to -\sqrt{2} \) from the right, \( f(x) \to +\infty \).

### Step 5: Find horizontal asymptotes
Next, we check the behavior of \( f(x) \) as \( x \to \infty \) or \( x \to -\infty \):
\[ f(x) = \frac{3}{2 - x^2} \to 0 \quad \text{as} \quad x \to \pm \infty \]
This indicates that \( f(x) \) approaches 0 but never actually reaches it.

### Step 6: Conclusion on the range
From the analysis:
- The function can take on all values from \( -\infty \) to 0 (not including 0) and from 0 to \( +\infty \) (not including 0).
Thus, the range of \( f(x) \) is:
\[ \text{Range} = (-\infty, 0) \cup (0, +\infty) \]

### Final Answer
The range of \( f(x) = \frac{3}{2 - x^2} \) is \( (-\infty, 0) \cup (0, +\infty) \).

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