Question

Inverse functions anyone?
How do i find the inverse of
f(x) = 4e^(x-2) - 11

Answer 1

To find the inverse, solve for x in terms of f(x).
\[f(x)=4e^{x-2}-11\]\[\frac{f(x)-11}{4}=e^{x-2}\]Take the natural log of both sides:\[\ln(\frac{f(x)-11}{4})=\ln(e^{x-2})\]The natural log "takes away" the exponential\[\ln(\frac{f(x)-11}{4})=x-2\]\[\ln(\frac{f(x)-11}{4})+2=x\]
Let me know if you're okay with the reasoning to there.

Answer 2

ok. i suppose getting to that second step is where i need to work on. where e^x-2 is by itself

Answer 3

from there i understand completely though

Answer 4

Okay. So let me expand on that part:

Answer 5

Subtract 11 from both sides:\[f(x)-11=4e^{x-2}\]
Divide both sides by 4:
\[\frac{f(x)-11}{4}=e^{x-2}\]

Answer 6

That part?

Answer 7

yes. i feel once that parts all sorted and we get to the ln's its simple

Answer 8

Okidoki. Do you know where to go from there?

Answer 9

i think... just why did 11 get subtracted and not added

Answer 10

Oh, whoops. That's because I did it wrong :D

Answer 11

Plus everywhere instead of minus

Answer 12

ahh ok that makes a lot more sense now hah. its still + 2 though right .. at the end

Answer 13

Yep.

Answer 14

it makes a lot more sense now. thanks a lot

Answer 15

Wait wait, not done yet.

Answer 16

oh . theres more expanding hm

Answer 17

We should be here:\[\ln(\frac{f(x)+11}{4})+2=x\]
Do you know what to do from here?

Answer 18

yes. this is a property i think

Answer 19

Nah, it's something else.

Answer 20

so ln(1/4x+11/4x)+2

Answer 21

Each way is equally valid.

The problem is that we're solving for the inverse function, but we don't have f-inverse anywhere in that equation. All you do now is "switch x and y," with f(x) here playing the part of y.

Answer 22

\[\ln(\frac{x+11}{4})+2=f^{-1}(x)\]