If `f(x)=(a-x)/(a+x)`, the domain of `f^(-1)(x)` contains

Options

A. `(-oo,oo)`
B. `(-oo,-1)`
C. `(-1,oo)`
D. `(0,oo)`

Explanation

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

### Step 1: Write the function
We start with the function:
\[ f(x) = \frac{a - x}{a + x} \]

### Step 2: Set \( f(x) = y \)
To find the inverse, we set:
\[ y = \frac{a - x}{a + x} \]

### Step 3: Solve for \( x \) in terms of \( y \)
Rearranging the equation to solve for \( x \):
1. Multiply both sides by \( a + x \):
\[ y(a + x) = a - x \]
2. Distributing \( y \):
\[ ay + xy = a - x \]
3. Rearranging gives:
\[ ay + xy + x = a \]
4. Factoring out \( x \):
\[ ay = a - x(1 + y) \]
5. Isolating \( x \):
\[ x(1 + y) = a - ay \]
6. Thus, we have:
\[ x = \frac{a(1 - y)}{1 + y} \]

### Step 4: Write the inverse function
Now we can express the inverse function:
\[ f^{-1}(x) = \frac{a(1 - x)}{1 + x} \]

### Step 5: Determine the domain of \( f^{-1}(x) \)
The inverse function \( f^{-1}(x) \) is defined as long as the denominator is not zero:
\[ 1 + x \neq 0 \implies x \neq -1 \]

### Step 6: Write the domain of \( f^{-1}(x) \)
The domain of \( f^{-1}(x) \) includes all real numbers except \( -1 \):
\[ \text{Domain of } f^{-1}(x) = (-\infty, -1) \cup (-1, \infty) \]

### Final Answer
The domain of \( f^{-1}(x) \) contains all real numbers except \( -1 \).

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