Question

Suppose \(A=\\{1,2,3\\} .\) Let \(f: A \rightarrow A\) be the function \(f=\\{(1,2),(2,2),(3,1)\\},\) and let \(g: A \rightarrow A\) be the function \(g=\\{(1,3),(2,1),(3,2)\\} .\) Find \(g \circ f\) and \(f \circ g\)

Short answer

The result of \(g \circ f\) is given as \(\{(1,1),(2,1),(3,3)\}\) and the result of \(f \circ g\) is given by \(\{(1,1),(2,2),(3,2)\}\).

Step-by-step solution

Find \(g \circ f\)

To find \(g \circ f\), start with the function \(f\), then apply the function \(g\) to the result of \(f\). For every \(x \in A\): 1) Find \(f(x)\), 2) Then find \(g(f(x))\). Hence, the result of \(g \circ f\) is \(g(f(1)), g(f(2)), g(f(3))\). Substitute with values to get \(g(2), g(2), g(1)\) which equals to \(\{(1,1),(2,1),(3,3)\}\).

Find \(f \circ g\)

To find \(f \circ g\), start with the function \(g\), then apply the function \(f\) to the result of \(g\). For every \(x \in A\): 1) Find \(g(x)\), 2) Then find \(f(g(x))\). Hence, the result of \(f \circ g\) is \(f(g(1)), f(g(2)), f(g(3))\). Substitute with values to get \(f(3), f(1), f(2)\) which equals to \(\{(1,1),(2,2),(3,2)\}\).