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# The function $$f(x) = \frac{cx}{2x+3}$$satisfies $f(f(x))=x$ for all real numbers $x\ne -\frac 32$. Find $c$.

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The function$$f(x) = \frac{cx}{2x+3}$$satisfies $f(f(x))=x$ for all real numbers $$x\ne -\frac 32$$. Find $c$.

michaelcai  Dec 5, 2017
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c=-3

michaelcai  Dec 5, 2017
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f (f(x) )   =

c  [ cx / (2x + 3) ]

________________         =     x

2 [ cx / (2x + 3)] + 3

[c^2x] / (2x + 3) ]

___________________     =    x

[ 2cx + 6x + 9] / (2x + 3)

c^2x    =  [2cx + 6x + 9 ] * x

c^2x  = 2cx^2 + 6x^2 + 9x

(2c + 6)x^2 + (9 - c^2)x  =  0

For this to be 0  for all  real x {except x = -3/2}   we need both

2c + 6  = 0       and    9 - c^2  = 0

So  using the second    c  = 3   or  c = -3

But when  c  =  3,   2c + 6  =  12

But  when  c   = -3      both  2c + 6  and 9 - c^2    will = 0

So....  c  = - 3

CPhill  Dec 5, 2017

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