Determine whether the function \(f: \mathbb{Z} \to \mathbb{Z}\) defined by \(f(x) = 2x + 3\) is surjective (onto).

Step 1 ：Step 1: A function \(f: A \to B\) is said to be surjective (onto) if for every \(b \in B\), there exists at least one \(a \in A\) such that \(f(a) = b\).

Step 2 ：Step 2: Here, \(A = B = \mathbb{Z}\) (the set of all integers). So for the function \(f(x) = 2x + 3\) to be surjective, we need to find an integer \(x\) such that \(2x + 3 = b\) for any integer \(b\).

Step 3 ：Step 3: Solving the equation \(2x + 3 = b\) for \(x\), we get \(x = \frac{b - 3}{2}\).

Step 4 ：Step 4: However, \(x\) is not an integer for every integer value of \(b\). For instance, if \(b = 2\), then \(x = \frac{2 - 3}{2} = -\frac{1}{2}\), which is not an integer.

Step 5 ：Step 5: Therefore, we cannot find an integer \(x\) such that \(f(x) = b\) for every integer \(b\), which means that the function \(f(x) = 2x + 3\) is not surjective.