If f (x) = x-1/x+1 then find the value of f (2x).

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) ]

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