Step 1 :The function is defined as \(f(x) = x + 3\) for \(x < -4\) and \(f(x) = x - 3\) for \(x \geq -4\).
Step 2 :For the first part of the function, as \(x\) approaches \(-\infty\), \(f(x)\) also approaches \(-\infty\). As \(x\) approaches \(-4\), \(f(x)\) approaches \(-1\).
Step 3 :For the second part of the function, as \(x\) approaches \(-4\), \(f(x)\) approaches \(-7\). As \(x\) approaches \(\infty\), \(f(x)\) also approaches \(\infty\).
Step 4 :Therefore, the range of the function is \((-\infty, -1] \cup [-7, \infty)\).
Step 5 :Final Answer: The range of the piecewise function is \(\boxed{(-\infty, -1] \cup [-7, \infty)}\).