If ` f(x) = (3x +2)/(5x -3)` then

Options

A. `f^(-1) (x) = f(x)`
B. ` f^(-1) (x) =- f(x)`
C. ` (f o f) (x) =-x `
D. ` f^(-1) (x) = -(1)/(19) f(x)`

Answer

To solve the problem, we need to find the inverse of the function \( f(x) = \frac{3x + 2}{5x - 3} \) and check if \( f^{-1}(x) = f(x) \) or \( f^{-1}(x) = -f(x) \).

### Step-by-Step Solution:

1. **Set \( f(x) \) equal to \( y \)**:
\[ y = \frac{3x + 2}{5x - 3} \]

2. **Cross-multiply to eliminate the fraction**:
\[ y(5x - 3) = 3x + 2 \]
This simplifies to:
\[ 5xy - 3y = 3x + 2 \]

3. **Rearrange the equation to isolate terms involving \( x \)**:
\[ 5xy - 3x = 3y + 2 \]

4. **Factor out \( x \) from the left side**:
\[ x(5y - 3) = 3y + 2 \]

5. **Solve for \( x \)**:
\[ x = \frac{3y + 2}{5y - 3} \]

6. **Now, replace \( y \) with \( x \) to find the inverse function**:
\[ f^{-1}(x) = \frac{3x + 2}{5x - 3} \]

7. **Check if \( f^{-1}(x) = f(x) \)**:
Since we found that:
\[ f^{-1}(x) = \frac{3x + 2}{5x - 3} = f(x) \]
This means \( f^{-1}(x) = f(x) \).

### Conclusion:
The first option \( f^{-1}(x) = f(x) \) is correct.