Question

Consider the functions \(f, g: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) defined as \(f(m, n)=\left(m n, m^{2}\right)\) and \(g(m, n)=(m+1, m+n)\). Find the formulas for \(g \circ f\) and \(f \circ g\).

Short answer

The functions \(g \circ f\) and \(f \circ g\) are given by \(g \circ f(m, n) = (mn + 1, mn + m^2)\) and \(f \circ g(m, n) = (m^2 + 2mn + n, m^2 + 2m + 1)\) respectively.

Step-by-step solution

Define the Functions

The two functions are defined as follows: \(f(m, n) = (mn, m^2)\) and \(g(m, n) = (m+1, m+n)\). These are functions in \(\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\).

Find \(g \circ f\)

When finding \(g \circ f\), we apply 'f' first, and then 'g'. The function 'f' produces (mn, m^2). This output becomes the input for 'g'. Thus: \(g(f(m, n)) = g(mn, m^2) = ((mn)+1, (mn) + (m^2)) = (mn + 1, mn + m^2)\).

Find \(f \circ g\)

When finding \(f \circ g\), we apply 'g' first, and then 'f'. The function 'g' produces (m+1, m+n). This output becomes the input for 'f'. Thus: \(f(g(m, n)) = f(m+1, m+n) = ((m+1)(m+n), (m+1)^2) = (m^2 + 2mn + n, m^2 + 2m + 1)\).