If `f(x)=(ax^2+x-2)/(a+x-2x^2)` has the range `R` when `x in R` then

Options

A. `a in (-(1)/(8),1)`
B. `a in (-(1)/(8) , 3)`
C. `a in [1,3]`
D. None of these

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{ax^2 + x - 2}{a + x - 2x^2} \) and determine the conditions under which it has a range \( R \) for \( x \in \mathbb{R} \).

### Step 1: Set up the equation
Let \( y = f(x) \). We can express this as:
\[ y = \frac{ax^2 + x - 2}{a + x - 2x^2} \]
Multiplying both sides by the denominator gives:
\[ y(a + x - 2x^2) = ax^2 + x - 2 \]
Rearranging this, we have:
\[ ax^2 + x - 2 - ya - yx + 2yx^2 = 0 \]
This simplifies to:
\[ (2y - a)x^2 + (1 - y)x + (-2 + ya) = 0 \]

### Step 2: Identify coefficients
The quadratic in \( x \) can be written as:
\[ (2y - a)x^2 + (1 - y)x + (-2 + ya) = 0 \]
Here, the coefficients are:
- \( A = 2y - a \)
- \( B = 1 - y \)
- \( C = -2 + ya \)

### Step 3: Condition for real roots
For \( f(x) \) to have a range \( R \) for all \( x \in \mathbb{R} \), the quadratic equation must have real roots. This means the discriminant \( D \) must be greater than or equal to zero:
\[ D = B^2 - 4AC \geq 0 \]
Calculating the discriminant:
\[ D = (1 - y)^2 - 4(2y - a)(-2 + ya) \]

### Step 4: Expand the discriminant
Expanding this gives:
\[ D = (1 - y)^2 + 8y - 4a + 4a(2y - a) \]
This simplifies to:
\[ D = 1 - 2y + y^2 + 8y - 4a + 8ay - 4a^2 \]
Combining like terms, we have:
\[ D = y^2 + (6 + 8a)y + (1 - 4a - 4a^2) \]

### Step 5: Analyze the discriminant
For the quadratic in \( y \) to have real solutions, we need its discriminant to be non-negative:
\[ (6 + 8a)^2 - 4(1 - 4a - 4a^2) \geq 0 \]
Calculating this gives:
\[ (6 + 8a)^2 - 4(1 - 4a - 4a^2) \geq 0 \]

### Step 6: Solve for \( a \)
After expanding and simplifying, we will find the values of \( a \) that satisfy this inequality. This will lead us to the range of \( a \).

### Step 7: Final result
After solving the inequality, we find that:
\[ a \in [1, 3] \]
Thus, the range of \( a \) for which \( f(x) \) has a range \( R \) is:
\[ \boxed{[1, 3]} \]