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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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Part I (spoiler) | Comment 1 of 4
Really?  Nobody has done this yet?  
Here's part 1, solving for x if f(x) = f(1/x)

Let x = y
Then 3y = f(y)+4f(1/y) = 5f(y) 

Let x = 1/y
Then 3/y =  f(1/y)+4f(y) = 5f(y)

So 3y = 3/y

So y*y = 1.

x = y = 1 or -1 (only two solutions).

In fact, if af(x) + bf(1/x) = cx  (where c is non-zero), then  1 and -1 are the still the only x values for which f(x) = f(1/x).

I'll let somebody else solve part 2 before I present my solution, but I wanted to get the ball rolling. 

Edited on October 25, 2010, 10:24 am

Edited on October 25, 2010, 10:27 am
  Posted by Steve Herman on 2010-10-25 08:50:30

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