Find all possible values of expressions `(2+x^2)/(4-x^2)`

Explanation

To find all possible values of the expression \( \frac{2 + x^2}{4 - x^2} \), we will determine the range of this expression step by step.

### Step 1: Set the expression equal to \( y \)
Let:
\[ y = \frac{2 + x^2}{4 - x^2} \]

### Step 2: Rearrange the equation
Multiply both sides by \( 4 - x^2 \) (assuming \( 4 - x^2 \neq 0 \)):
\[ y(4 - x^2) = 2 + x^2 \]
This simplifies to:
\[ 4y - yx^2 = 2 + x^2 \]

### Step 3: Collect all terms involving \( x^2 \)
Rearranging gives:
\[ yx^2 + x^2 = 4y - 2 \]
Factoring out \( x^2 \):
\[ x^2(y + 1) = 4y - 2 \]

### Step 4: Solve for \( x^2 \)
Thus, we can express \( x^2 \) as:
\[ x^2 = \frac{4y - 2}{y + 1} \]

### Step 5: Determine the conditions for \( x^2 \)
Since \( x^2 \) must be non-negative (i.e., \( x^2 \geq 0 \)), we need:
\[ \frac{4y - 2}{y + 1} \geq 0 \]

### Step 6: Analyze the inequality
The fraction \( \frac{4y - 2}{y + 1} \) is non-negative when both the numerator and denominator are either both positive or both negative.

1. **Numerator**: \( 4y - 2 \geq 0 \) implies \( y \geq \frac{1}{2} \).
2. **Denominator**: \( y + 1 > 0 \) implies \( y > -1 \).

### Step 7: Determine intervals
- For \( y \geq \frac{1}{2} \), the denominator \( y + 1 \) is always positive.
- For \( y < \frac{1}{2} \), we check \( 4y - 2 < 0 \) which gives \( y < \frac{1}{2} \).

Thus, the critical points are:
- \( y = \frac{1}{2} \) (where the numerator is zero).
- \( y = -1 \) (where the denominator is zero).

### Step 8: Combine results
From our analysis:
- The expression is non-negative when \( y \geq \frac{1}{2} \) and \( y > -1 \).
- Therefore, the valid range of \( y \) is:
\[ y \in \left[-1, \frac{1}{2}\right) \cup \left[\frac{1}{2}, \infty\right) \]

### Final Result
The possible values of the expression \( \frac{2 + x^2}{4 - x^2} \) are:
\[ (-\infty, -1) \cup \left[\frac{1}{2}, \infty\right) \]