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

2. f(2x) = 4f(x)

3. f(x^2) = x + 1

4. f(f(sqrt(x-1))) = f(sqrt(x))

Possible answers, if you want them:

**Spoiler:**

Also, here are some equations that I was curious about (I don't know of their solutions or even if they exist):

1. f'(x) = f(f(x))

2. f(e^x) = f(x) + 1

3. f(x^2) + f(2x) + f(1) = 0

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

I was just wondering because I thought it would be interesting to find functions which followed certain properties, such as when their argument is raised to the power of two, the function returns a value equivalent to twice its current value plus one or something of the like. Also, I apologize for I have little experience in abstract algebra or any other 'higher' subject areas, so there could be some things I might overlook (also for my lack of latex). Hopefully y'all can help.