If `f:R-{-(1)/(2)}toR-{(1)/(2)}` is defined by `f(x)=(x-3)/(2x+1)`, then `f^(-1)(x)` is

Options

A. `(x-3)/(2x-1)`
B. `(x+3)/(2x-1)`
C. `(x+3)/(2x+1)`
D. `(x+3)/(1-2x)`

Answer

To find the inverse of the function \( f(x) = \frac{x - 3}{2x + 1} \), we will follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
We start by letting \( y = f(x) \):
\[ y = \frac{x - 3}{2x + 1} \]

### Step 2: Cross-multiply to eliminate the fraction
Next, we cross-multiply to get rid of the fraction:
\[ y(2x + 1) = x - 3 \]
This simplifies to:
\[ 2xy + y = x - 3 \]

### Step 3: Rearrange the equation
Now, we rearrange the equation to isolate \( x \):
\[ 2xy - x = -3 - y \]
Factoring out \( x \) from the left side gives:
\[ x(2y - 1) = -3 - y \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{-3 - y}{2y - 1} \]

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

### Step 6: Simplify the expression
We can rewrite the expression for clarity:
\[ f^{-1}(x) = \frac{-3 - x}{2x - 1} \]

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