Composition function

[Jane1728]

Hello,

This is the question I need help with, it is about 'composition function':

(I'm sorry I did not post a picture of the question, but it is because my books are in a language other than English.)

Given that f (x + 1/x) = x^2 / x^4 +1, (x different than zero), find f(x).

How I tried to solve it:

for x>0:
(x + 1/x) > 1/ (x^2 + 1/x^2)

and for x<0:
(x + 1/x) < - [1 / (x^2 +1/x^2)]

I am not able to find the value of X. I found the answer for this question and f(x) is equal to 1 / (x^2 - 2) (x=< - 2 or x>=2).

Thank you, I would appreciate any help.

 

[Romsek]

is it

[MATH]f\left(x+\dfrac 1 x\right) = \dfrac{x^2}{x^4}+1[/MATH]
or

[MATH]f\left(x+\dfrac 1 x\right) = \dfrac{x^2}{x^4+1}[/MATH]

 

[Steven G]

Let u = x + 1/x. So now you have f(u). Now write the right hand side, whatever it is, in terms of u and be done.

 

[pka]

What if \(\large f(x)=\dfrac{1}{x^2-2}~?\)

 

[Jane1728]

is it

[MATH]f\left(x+\dfrac 1 x\right) = \dfrac{x^2}{x^4}+1[/MATH]
or

[MATH]f\left(x+\dfrac 1 x\right) = \dfrac{x^2}{x^4+1}[/MATH]

is it

[MATH]f\left(x+\dfrac 1 x\right) = \dfrac{x^2}{x^4}+1[/MATH]
or

[MATH]f\left(x+\dfrac 1 x\right) = \dfrac{x^2}{x^4+1}[/MATH]

The second one. I’m sorry I typed it wrong.

 

[Jane1728]

T

Let u = x + 1/x. So now you have f(u). Now write the right hand side, whatever it is, in terms of u and be done.

thank you for the reply!

 

[Romsek]

The second one. I’m sorry I typed it wrong.

post #4 applies then