We are given the function $f(x) = 2^{x(x-1)}$ and are tasked with showing that its inverse function, $f^{-1}(x)$, has the form:

$f^{-1}(x) = \frac{1}{2} \left( 1 + \sqrt{1 + 4 \log_2 x} \right)$

Step-by-step approach:

1. Express the given function:

$f(x) = 2^{x(x-1)}$ This is a double exponential function in terms of $x$.

2. To find the inverse, start by solving $f(x) = y$ for $x$:

We write: $y = 2^{x(x-1)}$ Take the logarithm base 2 on both sides: $\log_2(y) = x(x-1)$ This is a quadratic equation in terms of $x$: $x^2 - x - \log_2(y) = 0$

3. Solve the quadratic equation for $x$:

Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -1$, and $c = -\log_2(y)$, we get: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-\log_2(y))}}{2(1)}$ Simplifying: $x = \frac{1 \pm \sqrt{1 + 4\log_2(y)}}{2}$

4. Since $x \geq 1$, we take the positive branch of the square root:

$x = \frac{1 + \sqrt{1 + 4\log_2(y)}}{2}$

Thus, the inverse function $f^{-1}(y)$ is: $f^{-1}(y) = \frac{1}{2} \left( 1 + \sqrt{1 + 4 \log_2 y} \right)$

This shows that the given inverse expression is correct.

Would you like any further details on the steps?

Related Questions:

1. What is the domain of the function $f(x) = 2^{x(x-1)}$?
2. How do you verify if the inverse of a function is correct?
3. Can the function $f(x) = 2^{x(x-1)}$ be inverted for all real numbers?
4. What are the properties of logarithmic functions used in the inverse process?
5. How can the quadratic formula be derived for different coefficients?

Tip:

When solving for the inverse of functions, always ensure that you select the appropriate branch (positive or negative) based on the given domain of the original function.