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So... First off: Is it possible for a function to contain itself? For instance, here's something I've been messing around with today:

[tex]f(x) = a ( f(x) ) +x[/tex]

Now... If I change the notation from [tex]f(x)[/tex] to [tex]y[/tex] for clarity, I get this equation, which I can rearrange in the following way:

[tex]y = ay + x[/tex]

[tex]y - ay = x[/tex]

[tex]y(1-a) = x[/tex]

[tex]y = \frac{x}{1-a}[/tex]

And, returning the notation to its original form:

[tex]f(x) = \frac{x}{1-a}[/tex]

In which case, my result (without all of the work I posted above) would be that

[tex]f(x) = a( f(x) ) + x[/tex]

is identical to

[tex]f(x) = \frac{x}{1-a}[/tex]

Is this correct, or incorrect; is there some property of functions that makes what I've done wrong? If it is correct, is there a name for this sort of "self-containing" function?

Thanks in advance for your replies... ^^