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 can follow these steps:

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

### Step 2: Rearrange the equation
Rearranging gives:
\[ y(4 - x^2) = 2 + x^2 \]
Expanding this, we have:
\[ 4y - yx^2 = 2 + x^2 \]

### Step 3: Collect all \(x^2\) terms on one side
Rearranging the equation leads to:
\[ yx^2 + x^2 = 4y - 2 \]
Factoring out \(x^2\):
\[ x^2(y + 1) = 4y - 2 \]

### Step 4: Solve for \(x^2\)
From the previous equation, 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 set up the inequality:
\[ \frac{4y - 2}{y + 1} \geq 0 \]

### Step 6: Analyze the inequality
To solve the inequality, we need to find when 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 critical points
The critical points from the inequalities are:
- \(y = -1\) (denominator becomes zero, undefined)
- \(y = \frac{1}{2}\) (numerator becomes zero)

### Step 8: Test intervals
We can test intervals around the critical points:
- For \(y < -1\): Both numerator and denominator are negative, thus the expression is positive.
- For \(-1 < y < \frac{1}{2}\): The numerator is negative and the denominator is positive, thus the expression is negative.
- For \(y = \frac{1}{2}\): The expression equals zero.
- For \(y > \frac{1}{2}\): Both numerator and denominator are positive, thus the expression is positive.

### Step 9: Conclusion
From the analysis, we find that the possible values of \(y\) are:
\[ y \in (-\infty, -1) \cup \left[\frac{1}{2}, \infty\right) \]

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