Let [tex]f^{-1}(x)[/tex] be the inverse of
[tex]f(x) = 3 \log_3 (x + 3) + 1[/tex]
Then by definition of inverse function,
[tex]f\left( f^{-1}(x) \right) = 3 \log_3 \left(f^{-1}(x) + 3\right) + 1 = x[/tex]
Solve for the inverse :
[tex]3 \log_3 \left(f^{-1}(x) + 3\right) + 1 = x[/tex]
[tex]3 \log_3 \left(f^{-1}(x) + 3\right) = x - 1[/tex]
[tex]\log_3 \left(f^{-1}(x) + 3\right) = \dfrac{x - 1}3[/tex]
[tex]3^{\log_3 \left(f^{-1}(x) + 3\right)} = 3^{\frac{x-1}3}[/tex]
[tex]f^{-1}(x) + 3 = 3^{\frac{x-1}3}[/tex]
[tex]f^{-1}(x) = 3^{\frac{x-1}3} - 3[/tex]
