Question 995666

{{{g(x) = x -2}}}  ,  {{{f(x) = 3x - 2}}}  ,  

find {{{x}}} such that {{{g(g(x)) = f(f(x))}}}

{{{g(g(x)) = g(x -2)}}}

{{{g(g(x)) = (x -2)-2}}}

{{{g(g(x)) = x -2-2}}}

{{{g(g(x)) = x -4}}}


{{{f(f(x)) = f(3x - 2)}}}

{{{f(f(x)) = 3(3x - 2) - 2}}}

{{{f(f(x)) = 9x - 6 - 2}}}

{{{f(f(x)) = 9x - 8}}}


{{{g(g(x)) = f(f(x))}}}

{{{ x -4 = 9x - 8}}}

{{{ 8 -4 = 9x - x}}}

{{{ 4 = 8x}}}

{{{ 4/8=x}}}

{{{ highlight(x=1/2)}}}


check:

{{{g(x) = x -2}}}=>{{{g(x) = 1/2 -2}}}=>{{{g(x) = -3/2}}}

{{{g(g(x))=g(-3/2) }}} 

{{{g(g(x))=(-3/2)-2 }}} 

{{{g(g(x))=(-3/2)-4/2 }}} 

{{{g(g(x))=-7/2 }}} 


{{{f(x) = 3x - 2}}}

{{{f(x) = 3(1/2)- 2}}}

{{{f(x) = 3/2- 4/2}}}

{{{f(x) = - 1/2}}}

{{{ f(f(x))=f(- 1/2)}}}

{{{ f(f(x))=3(- 1/2)-2}}}

{{{ f(f(x))=- 3/2-4/2}}}

{{{ f(f(x))=- 7/2}}}

so, {{{g(g(x)) = f(f(x))}}} for {{{x=1/2}}}