SOLUTION: Please help me with this question: If f(x) = x^2-x-1 and f[g(x)]=4x^2 + 10x + 5, find g(x). Any help will be greatly appreciated!

Question 1104188: Please help me with this question: If f(x) = x^2-x-1 and f[g(x)]=4x^2 + 10x + 5, find g(x).
Any help will be greatly appreciated!
Found 2 solutions by math_helper, KMST:
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
I found it by trial and error, using hints from
I assumed
where a=2 because we need to end up with
and 2ab - a = 10 �> a(2b-1) = 10 (EDIT: because a=2, this implies b=3, missed that earlier)
Another hint/constraint is that
I found a=2 and b=3 works.

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Of course, this worked out because I guessed the proper form for g(x).
Maybe another tutor may have a more structured method. I'm often amazed at the talent base here.
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Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
There has to be a better way, but here it goes.
If f[g(x)] =
={ f[g(x)] } =
To find the inverse function of , ,
we exchange and in
to get and then solve for .
We get , .
That is really two functions,
and when working with real numbers,
it is only defined for ,
but we continue, to see what we can get.
If
.
So, is a solution.

NOTES:
1) There was no problem with being defined only for ,
because we are going to apply it to
.
2)
Would we get another solution out of ?
Let's check.
In that case, we would have
.
If ,
could be another .
Let's see.

No. It does not work.
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