Question

The function \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) defined by the formula \(f(m, n)=(5 m+4 n, 4 m+3 n)\) is bijective. Find its inverse.

Short answer

The inverse function is \(f^{-1}(x, y) = (3x - 2y, 2x - y)\)

Step-by-step solution

Understanding the function

In this problem, \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) is given as \(f(m, n) = (5m + 4n, 4m + 3n)\). As mentioned, it is a bijective function, which means each element of the domain \(\mathbb{Z} \times \mathbb{Z}\) corresponds to a unique element in the co-domain \(\mathbb{Z} \times \mathbb{Z}\). Because of this property, it can be inverted.

Writing system of equations

To find its inverse, we will first express \(m\) and \(n\) in terms of \(x\) and \(y\), where \(f^{-1}(x, y) = (m, n)\). Then, we will get the following system of equations where \(x = 5m + 4n\) and \(y = 4m + 3n\).

Calculating for m and n

Multiplying second equation by 5 and first by 4 and subtracting second from the first, this would yield \(m = 3x - 2y\). Similarly, multiplying first equation by 3 and second by -4, then adding the first to the second would yield \(n = 2x - y\). Thus, the inverse function \(f^{-1}(x, y)\) is \((3x - 2y, 2x - y)\).