Question 1209310
.
The function f satisfies {{{f(sqrt(2x - 1))}}} = {{{1/(2x - 1)}}}.
for all x not equal to 1/2.  Find f(2).
~~~~~~~~~~~~~~~~~~~~~~~



        The idea is to find  x  such that    {{{sqrt(2x-1)}}} = 2

        and then calculate  f(2)  using the given functional equation.


        Below is an implementation of this idea in steps.



<pre>
<U>Step 1.</U>  We want to find x from equation

             {{{sqrt(2x-1)}}} = 2.


         Do all necessary transformations as follow

             2x-1 = 2^2 = 4,

             2x = 4 + 1 = 5,

              x = 5/2.



<U>Step 2</U>.  According to the functional equation,

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

         Substitute here x = 5/2.  Remember that x is determined in a way that {{{sqrt(2x-1)}}} = 2.

         Therefore, you will get    

                 f(2) = {{{1/(2*(5/2)-1)}}} = {{{1/(5-1)}}} = {{{1/4}}}.


<U>ANSWER<</U>.  f(2) = {{{1/4}}}.
</pre>

At this point, &nbsp;the problem is solved completely.