Beginning Activity Beginning Activity 1: Constructing a New Function
Let \(A = \left\{ {a, b, c, d} \right\}\text{,}\) \(B = \left\{ {p, q, r} \right\}\text{,}\) and \(C = \left\{ {s, t, u, v} \right\}\text{.}\) The arrow diagram in FigureĀ 6.23 shows two functions: \(f\x A \to B\) and \(g\x B \to C\text{.}\)
Notice that if \(x \in A\text{,}\) then \(f(x) \in B\text{.}\) Since \(f(x) \in B\text{,}\) we can apply the function \(g\) to \(f(x)\text{,}\) and we obtain \(g(f(x))\text{,}\) which is an element of \(C\text{.}\)
Using this process, determine \(g(f(a))\text{,}\) \(g ( f(b) )\text{,}\) \(g ( f(c) )\text{,}\) and \(g ( f(d) )\text{.}\) Then explain how we can use this information to define a function from \(A\) to \(C\text{.}\)
