if f(x/x^2+1)=(x^2/x^4+1) then f(x)=?

Question

if f(x/x^2+1)=(x^2/x^4+1) then f(x)=?

Answer 1

f ( x / [x^2 + 1]) = x^2 / (x^4 + 1)

I have no clue how to solve this right now but I will by the end of the hour (until somebody else does).

15 minutes later: I still have no clue how to approach this. The answer is definitely not x^2, because if f(x) = x^2, then

f ( x / [x^2 + 1]) = x^2 / [x^4 + 2x + 1], which isn't our goal.

Edit, next day later:

Found out the process to solve it. All you have to do is determine the functions to get from one form to the other. A couple of people have already answered the question correctly, and I'm interested in getting the "how" instead of the "what" In this case, we want to get from

(x / [x^2 + 1]) to ( (x^2) / ([x^4 + 1] )

(x / [x^2 + 1]). Take the reciprocal of it.
([x^2 + 1] / x). Square it.
([x^4 + 2x^2 + 1] / x^2). Subtract 2 from it.

( {[x^4 + 2x^2 + 1] / x^2} - 2 )
( {[x^4 + 2x^2 + 1] / x^2} - 2(x^2)/(x^2) )
( [x^4 + 2x^2 + 1 - 2x^2] / [x^2] )
( [x^4 + 1] / [x^2] ). Take the reciprocal.
( [x^2] / [x^4 + 1] ).

Now, all we have to do is start from x and REPEAT the exact same process.

x. "Take the reciprocal of it."
1/x. "Square it."
1/[x^2]. "Subtract 2 from it."
(1/[x^2] - 2). "Take the reciprocal."

1 / { (1/[x^2] - 2) }

That should be our answer;

f(x) = 1 / { (1/[x^2] - 2) }

Since this is a complex fraction (fractions within a fraction), let's clean it up by multiplying top and bottom by x^2.

f(x) = [x^2] / [1 - 2x^2]

And that should be what f(x) is! Let's test f( x / [x^2 + 1] ).

f( x / [x^2 + 1] ) = { x / [x^2 + 1] }^2 / {1 - 2(x / [x^2 + 1])^2}

Square the fractions,

= { [x^2] / [x^4 + 2x^2 + 1] } / {1 - 2( [x^2] / [x^4 + 2x^2 + 1] ) }

Multiply top and bottom by [x^4 + 2x^2 + 1],

= { [x^2] / { [x^4 + 2x^2 + 1] - 2x^2 }
= [x^2] / [x^4 + 1]

Which is what our goal was.

Answer 2

F(X)=1/(x^2-2)

Answer 3

f(x) = 1/((1/x^2) -2)

Answer 4

f(x) = 1/((1/x^2) -2)

Answer 5

OK to get from x/x^2+1 to x^2/x^4+1 we can simplify

f(1/x+1)= 1/x^2+1 cancelling the X's.

So we need to find f(x)

So to transform 1/x+1 into x we need

so subtract 1 and invert.

I we to this on 1/x^2+1 we get x^2

so f(x)=x^2