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Question
I am really stuck on the following problem.  I just don't know how to set up a path for the solution.  Could you please guide me?

Given that f(x^2 + 1) = 2x + f(x) for all real x and f(0) = 7, find the value of f(5).

I looked at: f(x^2 + 1) = 2x + 7. But that leads me nowhere.  So, I solved x^2 +1 = 2x + 2x +1 for x.  I found that x = 0 and x = 4.  Well, that took me nowhere also.

Thanks,
Jimmy

Answer
For x^2 + 1 = 5, x^2 = 4, so x=2.
To find f(5), we get f(5) = 2*2 + f(2).

To find f(5), we need to look at x=1, for x^2 + 1 when x=1.  Note that 1^2 + 1 = 2.
This says that f(2) = 2*1 + f(1) = 2 + f(1).  This says that f(5) = 4 + f(2) = 4 + 2 + f(1).
So, that says that f(5) = 6 + f(1).

For x=0, x^1+1=1, so we have f(1) = 2*1 + f(0) = 2 + f(0).
Putting this into the last equation gives f(5) = 6 + 2 + f(0) = 8 + f(0).

Since we are given f(0) = 7, this says that since f(5) = 8 + f(0),
and f(0) = 7, that f(5) = 8 + 7.  Last I knew, that is the same as f(5) = 15.

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Scott A Wilson

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