How Do You Find g(x) When f(g(x)) Equals a Different Function?

[Gshandle]

Homework Statement

Find g(x) if f(x) = (3x-1)/(2x+5) and f(g(x)) = (x+9)/(12x-11)

Homework Equations

N/A, as far as I know

The Attempt at a Solution

I tried doing it as though g(x) = y and it turned out like this:
(3y-1)/(2y+5)=(x+9)/(12x-11)
I very quickly saw that that wouldn't work though, so I'm kind of lost. This is independent work in order for me to be able to go into pre-calc, so while I can ask a teacher it is difficult to find time, hence why I'm relying on you guys :) Help would be much appreciated!

 

[Vorde]

Think about it like this: you have f(x), and you have f(x) in the specific case when x=g(x), what you are looking for is what does x have to be in f(x) to give f(g(x)).

So, what does x have to be exchanged with in (3x-1)/(2x+5) to give (x+9)/(12x-11)

 

[Gshandle]

Well, I already got that part of the logic which is why I set g(x) equal to y, but it didn't work, so I don't understand how to get it to work.

 

[micromass]

Maybe you can find $f^{-1}(y)$??

That is, set

$$y=\frac{3x-1}{2x+5}$$

and try to find x in function of y.

 

[Vorde]

Ok, I just went through the math and its a bit of a long haul. Should be fairly easy but its tedious.

What you want to do is take your f(x) function, replace x with g(x), set it equal to your f(g(x)) function and then solve for g(x).