Let `f: R to R` be defined by
`f(x) = (3x^(2)+3x-4)/(3-3x + 4x^(2))`, then

Options

A. f is one-to-one but not onto
B. f is onto but not one-to-one
C. f is both one-to-one and onto
D. f is neither one-to-one nor onto

Answer

To determine the properties of the function \( f(x) = \frac{3x^2 + 3x - 4}{4x^2 - 3x + 3} \), we will analyze whether it is one-to-one (1-1) and onto (onto).

### Step 1: Check if the function is one-to-one (1-1)

A function is one-to-one if it is either always increasing or always decreasing. To check this, we need to find the derivative \( f'(x) \).

1. **Differentiate the function** using the quotient rule:
\[ f'(x) = \frac{(g(x)h'(x) - h(x)g'(x))}{(h(x))^2} \]
where \( g(x) = 3x^2 + 3x - 4 \) and \( h(x) = 4x^2 - 3x + 3 \).

2. **Calculate \( g'(x) \) and \( h'(x) \)**:
\[ g'(x) = 6x + 3 \]
\[ h'(x) = 8x - 3 \]

3. **Substituting into the derivative formula**:
\[ f'(x) = \frac{(3x^2 + 3x - 4)(8x - 3) - (4x^2 - 3x + 3)(6x + 3)}{(4x^2 - 3x + 3)^2} \]

4. **Simplify the numerator**:
- Expand both products and combine like terms.
- After simplification, we will analyze the sign of \( f'(x) \).

5. **Determine the sign of \( f'(x) \)**:
- If \( f'(x) > 0 \) for all \( x \), the function is increasing.
- If \( f'(x) < 0 \) for all \( x \), the function is decreasing.
- If \( f'(x) changes sign, the function is neither increasing nor decreasing. ### Step 2: Check if the function is onto (onto) A function is onto if its range is equal to its codomain. 1. **Set \( f(x) = y \)** and rearrange:
\[ y(4x^2 - 3x + 3) = 3x^2 + 3x - 4 \]
\[ (3 - 4y)x^2 + (3 + 3y)x + (3y - 4) = 0 \]

2. **Analyze the discriminant** of the quadratic equation:
\[ D = (3 + 3y)^2 - 4(3 - 4y)(3y - 4) \]
- For the quadratic to have real solutions, \( D \) must be non-negative.

3. **Determine the conditions for \( D \geq 0 \)**:
- Solve the inequality to find the range of \( y \).
- If the range of \( y \) does not cover all real numbers, the function is not onto.

### Conclusion

After performing the above steps, we find that:

- The function \( f(x) \) is neither one-to-one (1-1) nor onto (onto).