SOLUTION: How do i find the inverse function of: f(x)=ln√(e^x - 2) I started it like this: f(x)=y <=> y=ln√(e^x-2) <=> e^y = √(e^x-2) <=> e^2y = e^x - 2 <=> e^x = e^2y+2 <=> ???

Algebra ->  Functions -> SOLUTION: How do i find the inverse function of: f(x)=ln√(e^x - 2) I started it like this: f(x)=y <=> y=ln√(e^x-2) <=> e^y = √(e^x-2) <=> e^2y = e^x - 2 <=> e^x = e^2y+2 <=> ???       Log On


   

Question 1163140: How do i find the inverse function of: f(x)=ln√(e^x - 2)
I started it like this:
f(x)=y <=> y=ln√(e^x-2) <=> e^y = √(e^x-2) <=> e^2y = e^x - 2 <=> e^x = e^2y+2 <=> ???
But I couldn't solve it till the end so i would appreciate any kind of help!
Found 3 solutions by Edwin McCravy, Theo, greenestamps:
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!


You were almost there!

y+=+ln%28sqrt%28e%5Ex-2%29%29

e%5Ey+=+sqrt%28e%5Ex-2%29 

e%5E%282y%29+=+e%5Ex+-+2  

e%5Ex+=+e%5E%282y%29+%2B+2 

ln%28e%5Ex%29+=+ln%28e%5E%282y%29%2B2%29

x+=+ln%28e%5E%282y%29%2B2%29

Now interchange x and y

y+=+ln%28e%5E%282x%29%2B2%29

Replace y by f-1(x)

f-1(x) = ln(e2x+2)

Edwin

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
this was a beauty, but i think i have it, even though it's not what i would have thought.

normally, when you solve for the inverse function, you do the following:

replace y with x and x with y and then solve for y.

for example, consider y = ln(x)
replace x with y and y with x to get x = ln(y)
this is true if and only if y = e^x.
the inverse function of y = ln(x) is y = e^x. *****
that can be seen in the following graph.



we'll do the same procedure with your problem.
your equation is y = ln(sqrt(e^(x-2))
replace x with y and y with x to get x = ln(sqrt(e^(y-2))
this is true if and only if e^(y-2) = e^x
this is true if and only if y-2 = x
solve for y to get y = x + 2
your inverse equation is y = x + 2 *****
that can be seen in the following graph.



you will notice that both the graph of the original equation and the graph of the inverse equation are straight lines.

i figured if the original equation was a straight line and was shown in the form of a natural log function, then the inverse function, being a straight line, could also be shown in the form of a natural log funcion.

i ran out of time to explain this fully, but the results of my investigation was that the inverse function could also be shown in the form of a natural log function.

i got the following:

the original function is y = ln(sqrt(e^(x-2))
the inverse function is y = ln(e^(2x+2))
that can be shown in the following graph.




take it for what it's worth.
your inverse function is either:
y = 2x + 2, or:
y = ln(e^(2x+2))

both of these can be seen on the graphs.
the graphs show them to be reflections about the line y = x and (x,y) = (y,x).
both of these are indications of inverse functions.

i'm actually surprised i was able to see the relationship, but it's there, even it i can't explain if very well.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The answer from the other tutor demonstrates the standard way of finding the inverse of a function, which you were trying to do.

For many functions, it is easy to find an inverse by using the idea that an inverse function "un-does" what the function does.

Using that concept, the inverse function has to perform the opposite operations, and in the reverse order, compared to the original function.

The given function does the following to the input value:
(1) raise e to that power
(2) subtract 2
(3) take the square root
(4) take the natural log

The inverse function must then
(1) raise e to the power
(2) square it
(3) add 2
(4) take the natural log

That sequence of operations gives us the inverse function:

x --> e%5Ex --> %28e%5Ex%29%5E2+=+e%5E%282x%29 --> e%5E%282x%29%2B2 --> ln%28e%5E%282x%29%2B2%29