The function $f : [0,7] \to [0,70)$ where $f(x) = x^3 - 12x^2 + 45x$ is
Step-by-Step Solution
Key Concept: For a quadratic, find the vertex to determine the range; compare range with codomain to determine if onto or into.
Factoring: $f(x) = 3(x^2 - 8x + 15) = 3(x - 3)(x - 5)$. The derivative $f'(x) = 6x - 24 = 0$ gives $x = 4$ as a critical point. At $x = 4$, $f(4) = 48 - 96 + 45 = -3$ is the minimum. Since the parabola opens upward, the range is $[-3, \infty)$. With codomain $\mathbb{R}$, $f(x)$ is many-one (multiple inputs map to same output) and onto (range equals codomain when restricted appropriately).
Correct Answer: onto