If `f : R - {1} rarr R, f(x) = (x-3)/(x+1)`, then `f^(-1)` (x) equals

Options

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

Answer

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

### Step 1: Set the function equal to \( y \)
Let \( y = f(x) \):
\[ y = \frac{x-3}{x+1} \]

### Step 2: Solve for \( x \) in terms of \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \). Start by cross-multiplying:
\[ y(x + 1) = x - 3 \]
Expanding the left side:
\[ yx + y = x - 3 \]

### Step 3: Rearranging the equation
Now, we will rearrange the equation to isolate \( x \):
\[ yx - x = -3 - y \]
Factor out \( x \) from the left side:
\[ x(y - 1) = -3 - y \]

### Step 4: Solve for \( x \)
Now, divide both sides by \( (y - 1) \) (assuming \( y \neq 1 \)):
\[ x = \frac{-3 - y}{y - 1} \]

### Step 5: Simplify the expression
We can simplify this expression:
\[ x = \frac{-3 - y}{y - 1} = \frac{-(y + 3)}{y - 1} \]
This can be rewritten as:
\[ x = \frac{y + 3}{1 - y} \]

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

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

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