Let `f: R-{3/5}->R` be 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)/(9)f(x)`

Explanation

To solve the problem step by step, we need to analyze the function \( f(x) = \frac{3x + 2}{5x - 3} \) and find its inverse.

### Step 1: Define the function
We start with the function:
\[ f(x) = \frac{3x + 2}{5x - 3} \]

### Step 2: Set \( f(x) \) equal to \( y \)
Let’s set \( f(x) \) equal to \( y \):
\[ y = \frac{3x + 2}{5x - 3} \]

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

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ 5xy - 3x = 3y + 2 \]
Now, factor out \( x \) from the left side:
\[ x(5y - 3) = 3y + 2 \]

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

### Step 6: Replace \( y \) with \( x \)
To find the inverse function, we replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{3x + 2}{5x - 3} \]

### Step 7: Compare \( f(x) \) and \( f^{-1}(x) \)
We see that:
\[ f^{-1}(x) = f(x) \]
This means that the function is its own inverse.

### Conclusion
Thus, we conclude that:
\[ f^{-1}(x) = f(x) \]