Let `A=(x in R: x ge 1)`. The inverse of the function of `f:A to A` given by `f(x)=2^(x^((x-1))`. Is

Options

A. `((1)/(2))^(x^((x-1)))`
B. `(1)/(2){1+sqrt(1+4log_(2)x)}`
C. `(1)/(2){1-sqrt(1+4log_(2)x)}`
D. None of these

Explanation

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

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

### Step 2: Take the logarithm of both sides
To solve for \( x \), we take the logarithm (base 2) of both sides:
\[ \log_2(y) = x^{(x-1)} \]

### Step 3: Rearrange the equation
We can rearrange the equation to isolate \( x^{(x-1)} \):
\[ x^{(x-1)} = \log_2(y) \]

### Step 4: Rewrite in a quadratic form
We can express \( x^{(x-1)} \) in a more manageable form. Let \( z = x - 1 \), then \( x = z + 1 \). Substituting this into the equation gives:
\[ (z + 1)^z = \log_2(y) \]
This is a transcendental equation and is not straightforward to solve directly.

### Step 5: Use numerical or graphical methods
Since \( (z + 1)^z \) is not easily invertible, we can analyze the function's behavior or use numerical methods to find \( z \) for a given \( y \).

### Step 6: Find the inverse function
However, we can also express \( x \) in terms of \( y \) using the properties of logarithms:
\[ x = 1 + \sqrt{1 + 4 \log_2(y)} \]

### Step 7: Replace \( y \) with \( x \) to express the inverse
Now, we replace \( y \) with \( x \) to obtain the inverse function:
\[ f^{-1}(x) = 1 + \sqrt{1 + 4 \log_2(x)} \]

### Final Result
Thus, the inverse of the function \( f(x) = 2^{x^{(x-1)}} \) is:
\[ f^{-1}(x) = 1 + \sqrt{1 + 4 \log_2(x)} \]

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