SOLUTION: If y(x) = x / 1-x , show that 1/2 [y(x) + y(-x)] = y(x^2)

Algebra ->  Functions -> SOLUTION: If y(x) = x / 1-x , show that 1/2 [y(x) + y(-x)] = y(x^2)      Log On


   

Question 1015396: If y(x) = x / 1-x , show that 1/2 [y(x) + y(-x)] = y(x^2)
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
If y(x) = x / 1-x , show that 1/2 [y(x) + y(-x)] = y(x^2)
------------------------------------------------------------ 

y(x) = x%2F%281-x%29,

y(-x) = %28-x%29%2F%281-%28-x%29%29 = -x%2F%281%2Bx%29.

Therefore, 

y(x) + y(-x) = x%2F%281-x%29 - x%2F%281%2Bx%29 =    (now write it with the common denominator, which is (1-x)*(1+x) )

= %28x%2A%281%2Bx%29%29%2F%28%281-x%29%2A%281%2Bx%29%29 - %28x%2A%281-x%29%29%2F%28%281-x%29%2A%281%2Bx%29%29 = 

= %28x+%2B+x%5E2+-+x+%2B+x%5E2%29%2F%281-x%5E2%29 = %282x%5E2%29%2F%281-x%5E2%29.

Therefore, 

1%2F2.%28y%28x%29+%2B+y%28-x%29%29 = x%5E2%2F%281-x%5E2%29.

It is what has to be proved.