If `f:Rto(-oo,1)` such that `f(x)=1-2^(-x)` then `f^(-1)(x)` is

Options

A. `1+log_(2)(-x)`
B. `1-log_(2)(x)`
C. `log_(2)(1-x)`
D. `-log_(2)(1-x)`

Answer

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

### Step 1: Set up the equation for the inverse function
We start by setting \( y = f(x) \):
\[ y = 1 - 2^{-x} \]

### Step 2: Rearrange the equation to isolate \( 2^{-x} \)
To isolate \( 2^{-x} \), we can rearrange the equation:
\[ 2^{-x} = 1 - y \]

### Step 3: Take the logarithm of both sides
Next, we take the logarithm (base 2) of both sides:
\[ -x = \log_2(1 - y) \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = -\log_2(1 - y) \]

### Step 5: Replace \( y \) with \( x \) to express the inverse function
Since we want \( f^{-1}(x) \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = -\log_2(1 - x) \]

### Final Result
Thus, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = -\log_2(1 - x) \]

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