Question

What is the inverse function of `f(x)=3^x+2`?

Answer 1

`y=(ln(x-2))/ln(3)`

Explanation:

Set `" "y=3^x+2`

The `y` is called the 'dependant variable' and the `x` is called the 'independent variable'. That is because we may assign any value we choose to `x` and the value of `y` depends on what we assign to `x`

We need to make `x` the dependant variable.

I am assuming you know the shortcut methods of manipulation.

Move the 2 to the other side

`y-2=3^x`

Take logs of both sides ( I choose ln)

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

This is the same as:

`ln(y-2)=xln(3)`

Move `ln(3)` to the other side

`(ln(y-2))/ln(3)=x`

Where there is a `y` write `x` and where there is a `x` write `y`

`(ln(x-2))/ln(3)=y`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Notice that

`y=(ln(x-2))/ln(3)`

is a reflection about `y=x`

of `y=3^x+2`