Let `f : D -> R`, where D is the domain off. Find the inverse of `f(x) = 1 - 2^-x`

Explanation

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

### Step 1: Set the function equal to \( y \)
Let \( y = f(x) \):
\[ y = 1 - 2^{-x} \]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \). So we swap \( x \) and \( y \):
\[ x = 1 - 2^{-y} \]

### Step 3: Rearrange the equation to isolate \( 2^{-y} \)
Rearranging the equation gives:
\[ 2^{-y} = 1 - x \]

### Step 4: Rewrite \( 2^{-y} \) in terms of positive exponent
Taking the reciprocal, we have:
\[ \frac{1}{2^y} = 1 - x \]

### Step 5: Multiply both sides by \( 2^y \)
This leads to:
\[ 1 = (1 - x) \cdot 2^y \]

### Step 6: Isolate \( 2^y \)
Rearranging gives:
\[ 2^y = \frac{1}{1 - x} \]

### Step 7: Take the logarithm of both sides
Taking the logarithm (base 2) of both sides:
\[ y = \log_2\left(\frac{1}{1 - x}\right) \]

### Step 8: Write the inverse function
Thus, we can express the inverse function \( f^{-1}(x) \) as:
\[ f^{-1}(x) = \log_2\left(\frac{1}{1 - x}\right) \]

### Final Answer
The inverse of the function \( f(x) = 1 - 2^{-x} \) is:
\[ f^{-1}(x) = \log_2\left(\frac{1}{1 - x}\right) \]

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