2nd Iberoamerican 1987

Find f(x) such that f(x)²f( (1-x)/(1+x) ) = 64x for x not 0, ±1.

Solution

Put x = (1-y)/(1+y), then (1--x)/(1+x) = y, so f( (1-y)/(1+y) )² f(y) = 64(1-y)/(1+y). Hence f( (1-x)/(1+x) )² f(x) = 64(1-x)/(1+x). But f(x)⁴ f( (1-x)/(1+x) )² = 64²x², so f(x)³ = 64 x²(1 + x)/(1 - x). Hence f(x) = 4 ( x²(1 + x)/(1 - x) )^1/3.

2nd Ibero 1987

© John Scholes
jscholes@kalva.demon.co.uk
1 July 2002