# Find the inverse of the function below. Graph the function below and the inverse function.  Determine the domain, range, and asymptotes of the function below and the inverse function. Please show...

Find the inverse of the function below. Graph the function below and the inverse function.

Determine the domain, range, and asymptotes of the function below and the inverse function. Please show all of your work. Submit both of your graphs.

f(x)=6-ln(x+2)

hala718 | Certified Educator

f(x)= 6-ln(x+2)

First we will find the domain:

The domain are all real numbers such that f(x) is defined.

We know that ln(x+2) is defined for (x+2)>0

==> x > -2

Then the domain is `x in (-2,oo).`

`` Now we will find the inverse.

First we switch x and y

==>  x = 6- ln(y+2)

Now we will isolate y.

`==gt x - 6 = -ln(y+2) `

`==gt ln(y+2) = 6-x `

`==gt y+2= e^(6-x)`

`` `==gt y= e^((6-x)) - 2`

Then, the inverse function is:

`f^-1 (x)= e^((6-x)) - 2`

rcmath | Certified Educator

To find the inverse function you set y=6-ln(x+2) ->6-y=ln(x+2)

->e^(6-y)=x+2 -> x=e^(6-y)-2

Thus the inverse function is g(x)=e^(6-x)-2