The function `f:R rarr R` defined as `f(x)=(3x^2+3x-4)/(3+3x-4x^2)` is :
(a) One to one but not onto
(b) Onto but not one to one
(c) Both one to one and onto
(d)Neither one to one nor onto

Options

A. One ot one buty not onto
B. Onto but not one to one
C. Both one to one and onto
D. Neither one to one nor onto

Answer

To determine the nature of the function \( f(x) = \frac{3x^2 + 3x - 4}{3 + 3x - 4x^2} \), we need to analyze whether it is one-to-one (injective) and/or onto (surjective).

### Step 1: Set up the equation
Let \( y = f(x) \):
\[ y = \frac{3x^2 + 3x - 4}{3 + 3x - 4x^2} \]

### Step 2: Cross multiply to eliminate the fraction
Multiply both sides by the denominator:
\[ y(3 + 3x - 4x^2) = 3x^2 + 3x - 4 \]
Expanding this gives:
\[ 3y + 3xy - 4xy^2 = 3x^2 + 3x - 4 \]

### Step 3: Rearrange the equation
Rearranging the equation, we get:
\[ 4xy^2 - 3x^2 + (3y - 3)x + (3y + 4) = 0 \]

### Step 4: Analyze the quadratic in \( x \)
This is a quadratic equation in \( x \). For the function to be one-to-one, this quadratic must have at most one solution for each \( y \). We can determine this by checking the discriminant of the quadratic equation.

### Step 5: Calculate the discriminant
The discriminant \( D \) of a quadratic \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
In our case:
- \( a = 4y \)
- \( b = 3y - 3 \)
- \( c = 3y + 4 \)

Thus, the discriminant is:
\[ D = (3y - 3)^2 - 4(4y)(3y + 4) \]
Calculating this:
\[ D = 9y^2 - 18y + 9 - 48y^2 - 64y \]
\[ D = -39y^2 - 82y + 9 \]

### Step 6: Analyze the discriminant
For the function to be one-to-one, the discriminant must be less than or equal to zero:
\[ -39y^2 - 82y + 9 \leq 0 \]
This is a downward-opening parabola. We need to find the roots of the equation:
\[ 39y^2 + 82y - 9 = 0 \]
Using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-82 \pm \sqrt{82^2 - 4 \cdot 39 \cdot (-9)}}{2 \cdot 39} \]
Calculating the discriminant:
\[ D' = 82^2 + 4 \cdot 39 \cdot 9 = 6724 + 1404 = 8128 \]
Thus, the roots are:
\[ y = \frac{-82 \pm \sqrt{8128}}{78} \]
Since the discriminant is positive, there are two distinct real roots, indicating that the quadratic can change signs.

### Step 7: Conclusion about one-to-one
Since the discriminant of the quadratic in \( x \) can be positive for some values of \( y \), the function is **not one-to-one**.

### Step 8: Check onto
To check if the function is onto, we need to see if every real number \( y \) can be achieved by some \( x \). Since the discriminant can be zero or negative for some values of \( y \), it indicates that there are values of \( y \) that cannot be achieved, thus the function is **not onto**.

### Final Answer
The function \( f(x) \) is neither one-to-one nor onto.