This is a question on the test review packet I have for discrete mathematics.
Given: $f = \{(a, b), (b, a), (c, b)\}$ a function from $X = \{a, b, c\}$ to $X.$
(a) Write $f \circ f$ and $f \circ f \circ f$ as sets of ordered pairs.
(b) Define $f^n = f \circ f \circ \ldots \circ f$ to be the $n$-fold composition of $f$ with itself. Write $f^9$ and $f^{623}$ as sets of ordered pairs.
For part (a), I believe: $ f(a) = b\\ f(b) = a\\ f(c) = b\\ f(f) = \{(a, a), (b, b), (c, a)\} \\ f(f(f)) = \{(a, b), (b, a), (c, b)\}$
I think there is a pattern here with even exponents being equal to $ f(f) = \{(a, a), (b, b), (c, a)\}$ and odd being equal to $f(f(f)) = \{(a, b), (b, a), (c, b)\}.$
So how do we define $f^n = f \circ f \circ \ldots \circ f$ to be the $n$-fold composition of $f$ with itself?
Also I believe, $f^9$ and $f^{623}$ would both be $\{(a, b), (b, a), (c, b)\}.$