SOLUTION: The function f satisfies f(\sqrt{2x - 1}) = \frac{1}{2x - 1} for all x not equal to 1/2. Find f(2).

Algebra.Com
Question 1209310: The function f satisfies
f(\sqrt{2x - 1}) = \frac{1}{2x - 1}
for all x not equal to 1/2. Find f(2).
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.
The function f satisfies = .
for all x not equal to 1/2. Find f(2).
~~~~~~~~~~~~~~~~~~~~~~~


        The idea is to find  x  such that    = 2
        and then calculate  f(2)  using the given functional equation.

        Below is an implementation of this idea in steps. 


Step 1.  We want to find x from equation

              = 2.


         Do all necessary transformations as follow

             2x-1 = 2^2 = 4,

             2x = 4 + 1 = 5,

              x = 5/2.



Step 2.  According to the functional equation,

                  = .

         Substitute here x = 5/2.  Remember that x is determined in a way that  = 2.

         Therefore, you will get    

                 f(2) =  =  = .


ANSWER<.  f(2) = .

At this point,  the problem is solved completely.



Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

w = sqrt(2x-1)
f(w) = (1/w)^2
f(2) = (1/2)^2
f(2) = 1/4


Or,
sqrt(2x-1) = 2
2x-1 = 2^2
2x-1 = 4
2x = 5
x = 5/2 = 2.5
Then,
f( sqrt(2x-1) ) = 1/(2x-1)
f( sqrt(2*2.5-1) ) = 1/(2*2.5-1)
f(2) = 1/4
RELATED QUESTIONS

The function $f$ satisfies \[f(\sqrt{x + 1}) = \frac{1}{x}\]for all $x >= -1,$ $x is not (answered by ikleyn)
The function $f$ satisfies \[f(\sqrt{x + 1}) = \frac{1}{x}\]for all $x \ge -1,$ $x\neq... (answered by jim_thompson5910)
The function f satisfies f(sqrt(x+1))=1/x For all x >= -1, x does not equal 0. Find... (answered by ikleyn)
The function f(x) satisfies f(sqrt(x+1)) = 1/x for all x >= -1, x is not = 0. Find... (answered by robertb)
\[f(\sqrt{x + 1}) = \frac{1}{x}\] for all $x \ge -1,$ $x\neq 0.$ Find... (answered by MaxWong)
For all real numbers {{{x}}}, the function {{{f(x)}}} satisfies {{{2f(x) + f(1-x) =... (answered by Edwin McCravy)
simplify the expression {(f(x+h) - f(x)}/h, h not equal to 0 for f(x) = x squared -2x... (answered by josgarithmetic)
Let F(x) be the real-valued function defined for all real x except for x = 1 and x = 2... (answered by CPhill,ikleyn)
Find the inverse function of f f(x) = sqrt(2x-1) y=sqrt(2x-1) y^2 = 2x-1 y^2 + 1 =... (answered by stanbon)