Verified Answer :

arthurpdc

Firstly, we'll find y = f⁻¹(x):

\(f(x)=\log_3(x+1)\\\\ x=\log_3(y+1)\\\\ y+1=3^x\\\\ y=3^x-1\\\\ f^{-1}(x)=3^x-1\)

So:

\(f^{-1}(x)=3^x-1\\\\ f^{-1}(2)=3^2-1\\\\ f^{-1}(2)=9-1\\\\ \boxed{f^{-1}(2)=8}\)

Answer: The value is 8.

Step-by-step explanation:  Given function is

\(f(x)=\log_3(x+1).\)

we are given to find the value of \(f^{-1}(2).\)

Let us consider that

\(f(x)=y~~~~~\Rightarrow x=f^{-1}(y).\)

So, we have

\(f(x)=\log_3(x+1)\\\\\Rightarrow y=\log_3(f^{-1}(y)+1)\\\\\Rightarrow 3^y=f^{-1}(y)+1\\\\\Rightarrow f^{-1}(y)=3^y-1.\)

If y = 2, then we have

\(f^{-1}(2)=3^2-1=9-1=8.\)

Thus, the required value is 8.