Question:

Step 1: Set up the equation to find the inverse function. Let $y = f(x) = 2^{x(x-1)}$. To find the inverse function, we need to express $x$ in terms of $y$. Step 2: Take logarithm base 2 of both sides. Taking $\log_2$ of both sides of the equation: $$\log_2 y = \log_2(2^{x(x-1)})$$ $$\log_2 y = x(x-1)$$ $$\log_2 y = x^2 - x$$ Step 3: Rearrange into standard quadratic form. Rearranging the equation to standard quadratic form in terms of $x$: $$x^2 - x - \log_2 y = 0$$ Step 4: Apply the quadratic formula. Using the quadratic formula with $a = 1$, $b = -1$, and $c = -\log_2 y$: $$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-\log_2 y)}}{2(1)}$$ $$x = \frac{1 \pm \sqrt{1 + 4\log_2 y}}{2}$$ Step 5: Determine the correct sign using the domain constraint. Since the domain of $f(x)$ is $\left[\frac{1}{2}, \infty\right)$, we require $x \geq \frac{1}{2}$. Testing the two solutions: - With the positive sign: $x = \frac{1 + \sqrt{1 + 4\log_2 y}}{2} \geq \frac{1}{2}$ ✓ - With the negative sign: $x = \frac{1 - \sqrt{1 + 4\log_2 y}}{2}$ would give $x < \frac{1}{2}$ for the range of $y$ ✗ Therefore, we take the positive sign. Step 6: Write the inverse function. Replacing $y$ with $x$ to express the inverse function: $$f^{-1}(x) = \frac{1 + \sqrt{1 + 4\log_2 x}}{2}$$ $$f^{-1}(x) = \frac{1}{2}\left(1 + \sqrt{1 + 4\log_2 x}\right)$$ **Final Answer:** The inverse function is $f^{-1}(x) = \dfrac{1}{2}\left(1 + \sqrt{1 + 4\log_2 x}\right)$, which corresponds to **Option 1**.