If `f(x)=(x)/(x+1)`, then `f^(-1)(x)` is :

Options

A. `(x+1)/(x)`
B. `(x)/(1+x)`
C. `(x-1)/(x)`
D. none of these

Answer

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

### Step 1: Set up the equation
We start by letting \( y = f(x) \):
\[ y = \frac{x}{x + 1} \]

### Step 2: Solve for \( x \)
Next, we need to solve this equation for \( x \) in terms of \( y \). We can cross-multiply to eliminate the fraction:
\[ y(x + 1) = x \]
This simplifies to:
\[ yx + y = x \]

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

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

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

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

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