ƒ(x) = ( x – 1 ) / ( x + 1 ) ………… (1)
∴ ( x + 1 ). ƒ(x) = x – 1
∴ x. ƒ(x) + ƒ(x) = x – 1
∴ x. ƒ(x) – x = – 1 – ƒ(x)
∴ x. [ ƒ(x) – 1 ] = – [ 1 + ƒ(x) ]
∴ x = [ 1 + ƒ(x) ] / [ 1 – ƒ(x) ] ……………… (2)
∴ from (1),
ƒ(2x) = [ (2x) – 1 ] / [ (2x) + 1 ]
= { 2( [1+ƒ(x)] / [1-ƒ(x)] ) – 1 } / { 2( [1+ƒ(x)] / [1-ƒ(x)] + 1 }
= { 2[ 1 + ƒ(x) ] – [ 1 – ƒ(x) ] } / { 2[ 1 + ƒ(x) ] + [ 1 – ƒ(x) ] }
= { 2 + 2.ƒ(x) – 1 + ƒ(x) } / { 2 + 2.ƒ(x) + 1 – ƒ(x) }
= [ 1 + 3.ƒ(x) ] / [ 3 + ƒ(x) ]