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Question: Find the center of the hyperbola given by the equation
This is the standard form of a hyperbola centered at $ (1, 3) $.
We are given two functional equations:
Additivity: $ f(x + y) = f(x) + f(y) $
Multiplicativity: $ f(xy) = f(x)f(y) $
Thus, the only possible functions are $ f(x) = 0 $ and $ f(x) = x $.
$ f(x) = 0 $: clearly additive and multiplicative.
$ f(x) = x $: $ f(x + y) = x + y = f(x) + f(y) $, and $ f(xy) = xy = f(x)f(y) $.
Hence, there are exactly two such functions.
Question: Find the area of the region enclosed by the graph of