Relations & Functions
Inverse Functions
Grade 12
Question:
<p>If the function \( f(x) \) on the domain \( \left[\dfrac{1}{2}, \infty\right) \) is defined by \( f(x) = 2^{x(x-1)} \), then \( f^{-1}(x) \) equals:</p>
<p>(a) \( \dfrac{1}{2}\left(1 + \sqrt{1 + 4\log_2 x}\right) \)</p>
<p>(b) \( \dfrac{1}{2}\left(1 - \sqrt{1 + 4\log_2 x}\right) \)</p>
<p>(c) \( \sqrt{1 + 4\log_2 x} \)</p>
<p>(d) \( \sqrt{1 - 4\log_2 x} \)</p>
Step-by-Step Solution
Key Concept: To find the inverse function, set y = 2^(x(x-1)), take logarithm base 2, and solve the resulting quadratic equation for x in terms of y. The domain restriction ensures the quadratic has a valid solution that respects the original domain.
<p><strong>Step 1:</strong> Set y = f(x) = 2^(x(x-1))</p><p><strong>Step 2:</strong> Take log base 2 on both sides: log₂(y) = x(x-1) = x² - x</p><p><strong>Step 3:</strong> Rearrange into standard quadratic form: x² - x - log₂(y) = 0</p><p><strong>Step 4:</strong> Apply quadratic formula: x = (1 ± √(1 + 4log₂(y)))/2</p><p><strong>Step 5:</strong> Since domain is [1/2, ∞), we need x ≥ 1/2. The positive root satisfies this condition:</p><p>x = (1 + √(1 + 4log₂(y)))/2</p><p><strong>Step 6:</strong> Replace y with x to get the inverse function:</p><p>∴ f⁻¹(x) = (1 + √(1 + 4log₂(x)))/2</p>
Correct Answer: A