Beginning Activity Beginning Activity 2: A Composition of Two Specific Functions
Let \(A = \left\{ {a, b, c, d} \right\}\) and let \(B = \left\{ {p, q, r, s} \right\}\text{.}\)
1.
Construct an example of a function \(f\x A \to B\) that is a bijection. Draw an arrow diagram for this function.
2.
On your arrow diagram, draw an arrow from each element of \(B\) back to its corresponding element in \(A\text{.}\) Explain why this defines a function from \(B\) to \(A\text{.}\)
3.
If the name of the function in Exercise 2 is \(g\text{,}\) so that \(g\x B \to A\text{,}\) what are \(g( p )\text{,}\) \(g( q )\text{,}\) \(g( r )\text{,}\) and \(g( s )\text{?}\)
4.
Construct a table of values for each of the functions \(g \circ f\x A \to A\) and \(f \circ g\x B \to B\text{.}\) What do you observe about these tables of values?