Final Answer:
The domain of the function h(t) = -4.87t² + 18.754 is all real numbers, represented as t ∈ ℝ. The range of the function h(t) is h(t) ≤ 18.754 meters, where h(t) represents the height of the object projected in the air.
Step-by-step explanation:
The domain of a function refers to all possible input values for which the function is defined. In this case, the function h(t) = -4.87t² + 18.754 involves a quadratic term (-4.87t²), where t represents time in seconds. Since time can be any real number (positive, negative, or zero) in the context of this model, there are no restrictions on the values t can take. Therefore, the domain spans all real numbers (t ∈ ℝ).
Regarding the range, the function h(t) represents the height of an object projected in the air. The highest point the object reaches is given by the vertex of the parabolic function. The negative coefficient of the t² term (-4.87) implies an inverted parabola, indicating that the maximum value of h(t) occurs at the vertex, which is at t = 0. Plugging t = 0 into the function gives h(0) = 18.754 meters. Therefore, the range of the function is h(t) ≤ 18.754 meters, as the height of the object never exceeds this maximum value.
Understanding the nature of the parabolic function helps determine its behavior over time. As time progresses beyond t = 0, the height of the object decreases, reaching zero when it hits the ground. The maximum height reached is the initial height, and as the squared term in the function is negative, the height decreases over time until the object lands. This analysis confirms that the range of the function h(t) is bounded above by the initial height of 18.754 meters and extends downward from there as the object descends.