Question 109091
<pre><font size = 3><b>
Given h(x), find a pair of functions f and g such that h(x) = f<font face = "bookshelf symbol 7">g</font>g(x):
h(x) = square root {{{sqrt( (7x^7+1)/(7x^7-1) )  }}}. 

Pick g(x) as any expression in x from the whole expression

{{{sqrt( (7x^7+1)/(7x^7-1) )  }}}

such that if you were to substitute u for that expression,
there will be no x's left.

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Suppose we pick u to be {{{7x^7}}}, then we replace both
the {{{7x^7}}}'s by u and get

{{{sqrt( (u+1)/(u-1) )  }}}  

g(x) is simply equal to what we picked u to be, that is,
g(x) = {{{7x^7}}} 

To get f(x) we substitute x for u in

{{{sqrt( (u+1)/(u-1) )  }}} 

and get 

{{{sqrt( (x+1)/(x-1) )  }}}

So  

f(x) =  {{{sqrt( (x+1)/(x-1) )  }}}
g(x) =  {{{7x^7}}}

Checking:

h(x) = f<font face = "bookshelf symbol 7">g</font>g(x) = f(g(x)) = f({{{7x^7}}}) =  {{{sqrt( (7x^7+1)/(7x^7-1) )  }}}.

Edwin</pre>