Inverse Function

Question

So I'm studying for a final and one of the questions in the book is:

Let $A = \{1, 2, 3\}$. Define $f : A \to A$ by $f(1) = 2, f(2) = 1,$ and $f(3) = 3$.

Find $f^2 , f^3 , f^4$ and $f^{-1}$.

I understand how to find the inverse, but I'm not understanding the whole $f^2, f^3$ concept.

An explanation would be greatly appreciated.

Answer 1

2 things you need to know:

1. $f^2$ usally means $f\circ f$, while $f^3$ means $f\circ f\circ f$
2. for any two functions $g, h$ you should already know that $g\circ h$ is defined as $(g\circ h)(x)=g(h(x))$

Using these two things, you can calculate $f^2(x)$ for any $x$. For example, $f^2(2)$ is equal to $(f\circ f)(2)=f(f(2)) = f(1)=2$. You can do the same for the other two values of $x$.