Grade 11Algebra

If f (x) = x-1/x+1 then f (2x) is ? It must be in terms of f (x)

Arin Som

9 Years agoGrade 11

2 Answers

Sidhaarth Suresh

8 Years ago

∵ 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) ]

Rishi Sharma

5 Years ago

Dear Student,
Please find below the solution to your problem.

∵ 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) ]

Thanks and Regards