Question

Find the inverse function of the following=
f(x)=2∙3^x+9^x

1.   log₃⁡((√(x+1))+1)
2.   log₃⁡((√(x+1))-1)
3.   log₃⁡((√(x-1))+1)
4.   log₃⁡((√(x+1))-2)

Comments

can someone solve this using Log??

Answer 1

Let 3^x  = t and f(x) = y, then the given function can be rewritten as :

       y = 2t + t²

⇒    y =  t²  + 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   =  log₃(√ (y + 1) - 1)

Hence now to find inverse , we swap x and y , 

⇒   y   =  log₃((√ (x + 1) ) - 1)

Hence 2) is the correct answer..