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The derivative test is a helpful tool in understanding how a function behaves across its domain. It allows us to conclude whether a function is constantly increasing or decreasing by analyzing its derivative. For the function given, \( f(x) = e^{x^3 - 3x + 2} \), the derivative is calculated as: \[ f'(x) = e^{x^3 - 3x + 2} \cdot (3x^2 - 3) \] This derivative is crucial in investigating the function's monotonicity. A derivative test checks if it is positive or negative across the domain. If \( f'(x) > 0 \) throughout the domain \((-\infty, -1]\), then the function is strictly increasing in this interval. Conversely, if \( f'(x) < 0 \), the function would decrease. In this exercise, the derivative is always positive within the domain, hinting at a one-one nature of the function.

Monotonicity refers to whether a function is consistently increasing or decreasing. This characteristic is significant because it helps determine if a function is one-one or many-one. For the present function \( f(x) = e^{x^3 - 3x + 2}\), we derived that:\[ f'(x) = e^{x^3 - 3x + 2} \cdot (3x^2 - 3) \] To evaluate this, consider the simplifying term \(3(x^2 - 1)\). Within the domain \((-\infty, -1]\), \(x^2 > 1\). Therefore, \(3(x^2 - 1) > 0\). Since the exponential function \(e^{x^3 - 3x + 2}\) is always positive, the product is positive, confirming the function is strictly increasing across the given domain. Knowing this, we understand these increasing functions are also injective or one-one.

An increasing function is one whose output grows as the input value increases. For functions like \(f(x) = e^{x^3 - 3x + 2}\), which are strictly increasing on their interval, each input produces a unique output. This is clear from the derivative test: - Since \( f'(x) > 0 \) for all \(x\) in the domain \((-\infty, -1]\), the function is said to be strictly increasing.- Such functions are injective, meaning they assign a unique output to each input, ensuring there are no repeated values in the range.By understanding whether a function is increasing, one can categorize it as one-one, which is essential in this exercise for establishing the nature of the function.

Understanding the domain and range is essential in grasping a function's capabilities and limitations. The domain refers to all possible input values, while the range indicates all potential outputs. For this function, \(f(x) = e^{x^3 - 3x + 2}\), the stated domain is \((-\infty, -1]\), and the intended range is \((0, e^5]\).- **Domain:** Informs which \(x\) values are permissible for evaluating the function. Here it includes all numbers less than or equal to \(-1\).- **Range:** Describes the values that \(f(x)\) can assume. Through calculation, it achieves values more than 0 up to, but not surpassing, \(e^5\).To check if a function is onto, we investigate whether every value in the range can be paired with a corresponding \(x\). In this example, since outputs achieve values close to but not equaling the maximum \(e^5\), the function fits the description of a function that is into rather than onto. This distinction is crucial for correctly interpreting the function's nature.