If f(x) = `(x^2 -2x + 4)/(x^2+2x+4), x in R` then range of function is

Options

A. `[1//3, 3]`
B. `(1//3, 3)`
C. (3, 3)
D. `(-1//3, 3)`

Answer

To find the range of the function \( f(x) = \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \), we can follow these steps:

### Step 1: Set the function equal to \( y \)
Let \( f(x) = y \). Therefore, we have:
\[ y = \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ y(x^2 + 2x + 4) = x^2 - 2x + 4 \]
This simplifies to:
\[ yx^2 + 2yx + 4y = x^2 - 2x + 4 \]

### Step 3: Rearrange the equation
Rearranging the equation to one side results in:
\[ yx^2 - x^2 + 2yx + 2x + 4y - 4 = 0 \]
This can be simplified to:
\[ (y - 1)x^2 + (2y + 2)x + (4y - 4) = 0 \]

### Step 4: Identify coefficients
From the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = y - 1 \)
- \( b = 2y + 2 \)
- \( c = 4y - 4 \)

### Step 5: Use the discriminant condition
For \( x \) to have real solutions, the discriminant \( D \) must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]
Substituting the coefficients:
\[ (2y + 2)^2 - 4(y - 1)(4y - 4) \geq 0 \]

### Step 6: Expand the discriminant
Expanding gives:
\[ 4y^2 + 8y + 4 - 4[(y - 1)(4y - 4)] \geq 0 \]
Calculating \( (y - 1)(4y - 4) \):
\[ = 4y^2 - 4y - 4y + 4 = 4y^2 - 8y + 4 \]
Thus, we have:
\[ 4y^2 + 8y + 4 - 4(4y^2 - 8y + 4) \geq 0 \]
This simplifies to:
\[ 4y^2 + 8y + 4 - 16y^2 + 32y - 16 \geq 0 \]
Combining like terms:
\[ -12y^2 + 40y - 12 \geq 0 \]

### Step 7: Simplify the inequality
Dividing the entire inequality by -4 (and reversing the inequality sign):
\[ 3y^2 - 10y + 3 \leq 0 \]

### Step 8: Factor the quadratic
Factoring gives:
\[ (3y - 1)(y - 3) \leq 0 \]

### Step 9: Find the critical points
The critical points are:
\[ y = \frac{1}{3}, \quad y = 3 \]

### Step 10: Test intervals
Testing the intervals:
- For \( y < \frac{1}{3} \): Positive
- For \( \frac{1}{3} < y < 3 \): Negative
- For \( y > 3 \): Positive

Thus, the solution to the inequality is:
\[ \frac{1}{3} \leq y \leq 3 \]

### Conclusion
The range of the function \( f(x) \) is:
\[ \boxed{\left[\frac{1}{3}, 3\right]} \]