Question

Calculate the inverse function and the domain of the functions:
f(x)= 3 +ln(e^x- 2)

So far this is my working. I'm not sure of what to do with the -2

y=3+ln(e^x-2)
y-3=ln(e^x-2)

Answer 1

so far this is my working

f(x)= 3 +ln(e^x- 2)
y=3+ln(e^x-2)
y-3=ln(e^x-2)
e^(y-3)=e^x-2
2+e^(y-3)=e^x
2+y-3=ln(e^x )
y-1=x
y=x+1

Answer 2

which grade are you?

Answer 3

let h(x) = ln (e^x -2)
g(x) = x +3
so that, you have f(x) = g(h(x)
--> \(f^{-1}(x) = (g(h(x))^{-1} = h^{-1}(g^{-1}(x))\)
Now, find out \(g^{-1}(x) = x-3\)
replace into the expression above, you have
\(f^{-1}(x) = h^{-1}(x-3)\)
finding \(h^{-1}(x-3)\)

Answer 4

\(h(x) = ln (e^x +2)\\h(x-3) = ln (e^{x-3}+2)\)
inverse of this guy is the required function.

Answer 5

Apply "switch the role" rule, you have
\(e^{x-3} = ln (h^{-1}(x-3) +2)\)
e both sides.... can you step up??
This problem is not a piece of cake, that's why I asked "what grade are you?". Hopefully, you are a senior in university to get what I mean.

Answer 6

I'm first year university. This problem has me going. Thanks for the help :)