If `A=R-{(3)/(5)}` and function `f:AtoA` is defined by `f(x)=(3x+2)/(5x-3)`, then

Options

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

Answer

To solve the problem, we need to find the inverse of the function \( f(x) = \frac{3x + 2}{5x - 3} \) defined on the set \( A = \mathbb{R} - \left\{ \frac{3}{5} \right\} \).

### Step-by-Step Solution:

1. **Set up the equation for the function**:
We start by setting \( f(x) = y \):
\[ y = \frac{3x + 2}{5x - 3} \]

2. **Cross-multiply to eliminate the fraction**:
Multiply both sides by \( 5x - 3 \):
\[ y(5x - 3) = 3x + 2 \]

3. **Expand the left side**:
Distributing \( y \):
\[ 5xy - 3y = 3x + 2 \]

4. **Rearrange the equation**:
Move all terms involving \( x \) to one side and constant terms to the other side:
\[ 5xy - 3x = 3y + 2 \]

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

6. **Solve for \( x \)**:
Divide both sides by \( 5y - 3 \) (assuming \( y \neq \frac{3}{5} \)):
\[ x = \frac{3y + 2}{5y - 3} \]

7. **Write the inverse function**:
Thus, the inverse function is:
\[ f^{-1}(y) = \frac{3y + 2}{5y - 3} \]
Replacing \( y \) with \( x \):
\[ f^{-1}(x) = \frac{3x + 2}{5x - 3} \]

8. **Conclusion**:
We find that \( f^{-1}(x) = f(x) \), which means the function is its own inverse.

### Final Answer:
The function \( f(x) = \frac{3x + 2}{5x - 3} \) is its own inverse.