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.
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`
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