If `f" R -{3/5} to R` be defined by `f(x)=(3x+2)/(5x-3)`, then :

Options

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

Answer

To solve the problem, we need to find the inverse of the function \( f(x) = \frac{3x + 2}{5x - 3} \). Let's go through the steps to find \( f^{-1}(x) \).

### Step 1: Set up the equation
Start by letting \( y = f(x) \):
\[ y = \frac{3x + 2}{5x - 3} \]

### Step 2: Cross-multiply
To eliminate the fraction, cross-multiply:
\[ y(5x - 3) = 3x + 2 \]
This simplifies to:
\[ 5xy - 3y = 3x + 2 \]

### Step 3: Rearrange the equation
Rearranging the equation to isolate terms involving \( x \):
\[ 5xy - 3x = 3y + 2 \]
Factor out \( x \) from the left side:
\[ x(5y - 3) = 3y + 2 \]

### Step 4: Solve for \( x \)
Now, solve for \( x \):
\[ x = \frac{3y + 2}{5y - 3} \]

### Step 5: Write the inverse function
Since we have expressed \( x \) in terms of \( y \), we can write the inverse function:
\[ f^{-1}(y) = \frac{3y + 2}{5y - 3} \]
To express it in terms of \( x \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{3x + 2}{5x - 3} \]

### Conclusion
Thus, the inverse function is:
\[ f^{-1}(x) = \frac{3x + 2}{5x - 3} \]