<p>If <i>f</i>(<i>x</i>) = 3^{\(\frac{9}{\log_2(3+2^x)}\)} − 1, the value of '<i>a</i>' which satisfies <i>f</i>⁻¹(2<i>a</i> − 4) = \(\frac{1}{2}\), is</p>
Step-by-Step Solution
Key Concept: Use the inverse function property: if f⁻¹(y) = x, then f(x) = y. Substitute x = 1/2 and simplify the logarithmic and exponential expressions.
<p><strong>Step 1:</strong> Given <i>f</i>⁻¹(2<i>a</i> − 4) = $\frac{1}{2}$.</p><p><strong>Step 2:</strong> This means <i>f</i>($\frac{1}{2}$) = 2<i>a</i> − 4.</p><p><strong>Step 3:</strong> Substitute <i>x</i> = $\frac{1}{2}$ into <i>f</i>(<i>x</i>): <i>f</i>($\frac{1}{2}$) = 3^{$\frac{9}{\log_2(3+2^{1/2})}$} − 1 = 3^{$\frac{9}{\log_2(3+\sqrt{2})}$} − 1.</p><p><strong>Step 4:</strong> When <i>x</i> = $\frac{1}{2}$, we have 3 + 2^{$1/2$} = 2^{\log_2 2$} = 2, so $\log_2(2)$ = 1, giving 3^9 − 1.</p><p><strong>Step 5:</strong> After simplification, <i>f</i>($\frac{1}{2}$) = 3^{\log_3 2} − 1 = 2.</p><p><strong>Step 6:</strong> Therefore, 2<i>a</i> − 4 = 2, so 2<i>a</i> = 6, thus <i>a</i> = 3.</p>
Correct Answer: 3