Question

Inverse of f(x)= 2^3x + 1

Answer 1

to find inverse
replace f(x) with y
solve for x
switch x and y and replace y with f(x)inverse or f⁻¹(x)

y=2^(3x)+1
subtract 1
y-1=2^(3x)
take log base 2 of both sides
log₂(y-1)=log₂(2^(3x))
move exponent (2x) to front
log₂(y-1)=3xlog₂(2)

remember that logₓ(x)=1 so
log₂(y-1)=3x(1)
log₂(y-1)=3x
divide both sides by 3

`( log_(2)(y-1) )/(3)`=x
switch x and y and replace y with f⁻¹(x)

f⁻¹(x)=
`( log_(2)(x-1) )/(3)`

Answer 2

To find the inverse of a function:
1. Replace f(x) with y.
2. Switch x and y.
3. Solve for y.
4. Replace y with f-1(x) (notation for inverse fucntions)


`f(x)=2^(3x)+1`
Let's use y instead of f(x).

`y=2^(3x)+1`
Switch the variables...

`x=2^(3y)+1`
Now, we solve for y.

`x-1=2^(3y)`

`3y =log_2(x-1)`

`y=(log_2(x-1))/(3)`
Now, just put that inverse function notation on there.


`\boxed{f^(-1)(x)=(log_2(x-1))/(3)}`