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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. This is where I am stuck.

Thanks,

Jimmy

For f(x^2 + 1) = 2x + f(x), we shall consider a series of substitutions to help us out.

substituting in x=0, we have

f(1) = 2* 0 + f(0) = 7

substituting in x=1,

f(1^2 + 1) = f(2) = 2 * 1 + f(1) = 2 + 7 = 9 ( it is known previously that f(1) = 7 )

substituting in x=2,

f(2^2 + 1) = f(5) = 2* 2 + f(2) = 4 + 9 ( it is known previously that f(2) = 9 )

= 13 (shown)

Hope this helps. Peace.

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