Inverse of functions

Question

A)Let F(x)=e^x+3 find the inverse of the function

 

b)let f(x)=log₃√x ,for X >0

show that f^-1(x)=3^2x

Answer 1

A) To find the inverse function replace F(x) with x, and x with F^-1(x) in the original equation.  This gives:

x=e ^F-1(x)+3

Then solve for F^-1(x) by first taking the natural long (ln) of 

both sides:

ln(x)=F^-1(x) + 3

or the inverse function is:

F^-1(x) = ln(x) - 3

 

B) In order to solve this, do the same thing:

 

x=log₃(sqrt(f^-1(x)))

 

The sqrt = "raise to the 1/2 power".  Recall that log(a^b) = b*log(a)

 

Then x = 1/2 * log₃(f^-1(x))

2x = log₃(f^-1(x))

 

Take inverse log of both sides:

 

3^2x = f^-1(x)