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Defined Function (Posted on 2010-10-24) Difficulty: 3 of 5
A function f, defined for all non-zero real numbers x, satisfies:

f(x) + 4f(1/x) = 3x.

[1] Find all values of x for which f(x)=f(1/x);

[2] Find all values of x for which f(x)=f(-x).

No Solution Yet Submitted by Jer    
Rating: 4.0000 (1 votes)

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Solution Part 2: Spoiler | Comment 2 of 4 |
Well, I can't let a good problem go unsolved

(1) Let x = y
Then  f(y)+4f(1/y) = 3y

(2) Let x = 1/y
Then  f(1/y)+4f(y) = 3/y
        
(3) Multiplying by 4, gives
  4f(1/y) + 16f(y) = 12/y

(4) Subtracting equation 1 from equation 3 gives
    15f(y) = 12/y - 3y

(5) Dividing by 15 gives f(y) as
      f(y) = 4/5y - y/5

(6) It is interesting to me that we did not need to calculate the function to solve part 1, but we do need it for part 2.

(7) At any rate, if f(x) = f(-x),
    then 4/5x - x/5 = -4/5x + x/5

    8/5x = 2x/5

Multiplying by 5x gives 8 = 2x*x
x = +2 or -2, as the only answer to part 2.

(8) Checking the solution.
     f(2) = 4/10 - 2/5 = 0
     f(-2) = -4/10 + 2/5 = 0, so f(2) = f(-2)

(9) Still checking
    F(1/2) =  8/5 - 1/10 = 15/10 = 3/2
    F(2) + 4f(1/2) = 6 = 3*2, so our basic definition is satisfied.

Nice problem, Jer!

   


 



  Posted by Steve Herman on 2010-10-25 21:52:25
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