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

Options

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

Answer

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

### Step 1: Set \( f(x) \) equal to \( y \)
Let \( y = f(x) = \frac{1-x}{1+x} \).

### Step 2: Rearrange the equation to solve for \( x \)
We can rearrange the equation to express \( x \) in terms of \( y \):
\[ y(1 + x) = 1 - x \]
Expanding this gives:
\[ y + yx = 1 - x \]

### Step 3: Collect all terms involving \( x \) on one side
Rearranging the equation, we get:
\[ yx + x = 1 - y \]
Factoring out \( x \) from the left side:
\[ x(y + 1) = 1 - y \]

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

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

### Step 6: Replace \( y \) with \( x \)
To express the inverse function in standard form, we replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{1 - x}{1 + x} \]

Thus, the final answer is:
\[ f^{-1}(x) = \frac{1 - x}{1 + x} \]

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