Let 3^{x } = t and f(x) = y, then the given function can be rewritten as :
y = 2t + t^{2}
⇒ y = t^{2 }+ 2t + 1 - 1
⇒ y = (t + 1)^{2} - 1
⇒ (t + 1)^{2 }= 1 + y
⇒ t = √ (y + 1) - 1
⇒ 3^{x} = √ (y + 1) - 1
Now taking log both sides , we have :
⇒ x = log_{3}(√ (y + 1) - 1)
Hence now to find inverse , we swap x and y ,
⇒ y = log_{3}((√ (x + 1) ) - 1)
Hence 2) is the correct answer..