91Ó°ÊÓ

First, replace the function notation, \(f(x)\), with \(y\). We'll do this to make it easier to manipulate the equation and swap variables later. $$ y = 5 + 3 \log_6 (2x + 1) $$

Now, swap all \(x\) and \(y\) in the equation. Doing this allows us to isolate \(y\) in the next step and find the inverse function. $$ x = 5 + 3 \log_6 (2y + 1) $$

To find the inverse function, we now need to solve the equation for \(y\). To do this, we'll perform the following steps: - First, isolate the logarithmic term by subtracting 5 from both sides of the equation. - Then, divide by 3 to isolate the logarithm further. - Next, use the logarithmic properties to remove the log term. - Finally, isolate \(y\) by solving for it. $$ x - 5 = 3 \log_6 (2y + 1) $$ $$ \frac{x - 5}{3} = \log_6 (2y + 1) $$ We know the logarithmic property, \(b^x = a \Leftrightarrow x = \log_b a\). Here, \(b = 6\). Thus, we have: $$ 6^{\frac{x - 5}{3}} = 2y + 1 $$ Solve for \(y\): $$ 2y = 6^{\frac{x - 5}{3}} - 1 $$ $$ y = \frac{6^{\frac{x - 5}{3}} - 1}{2} $$ Finally, replace \(y\) with the inverse function notation, \(f^{-1}(x)\): $$ f^{-1}(x) = \frac{6^{\frac{x - 5}{3}} - 1}{2} $$ Thus, we have found the inverse function, \(f^{-1}(x)\).