Given \(f ( x ) = 2 x^2 + 8 x + 1\). The parabola opens upwards, since \(a = 2\), which is a positive number. The vertex of the parabola has \(x\)-coordinate \(-\dfrac{b}{2a}=-2\). The \(y\)-coordinate of the vertex is \(f ( -2 ) = -7\). Thus, \(-7\) is the smallest output of the function. The range of the function is \([-7,\infty)\).