1. The problem is about understanding the domain and range of a function modeling a real-world scenario, specifically the height of a bottle rocket over time given by $$h(t) = -16t^2 + 64t + 3$$. 2. The formula for height is a quadratic function, which typically has a parabolic graph. In real-world contexts like this, the domain (possible values of $t$) is restricted to times when the rocket is in the air, usually starting at $t=0$ (launch time) and ending when the rocket hits the ground (height zero). 3. Important rule: In real-world problems, negative time or negative quantities often don't make sense, so the domain is restricted to non-negative values or values that fit the scenario. 4. However, if a problem involves scenarios like cliffs or other contexts where negative values of the independent variable (like position or time) are meaningful, then the domain can include negative values. The key is to interpret the variable and the context carefully. 5. For example, if $t$ represents time since launch, negative $t$ doesn't make sense. But if $x$ represents horizontal position relative to a cliff edge, negative $x$ values could represent positions on one side of the cliff. 6. So, when you face a question with negative values, first understand what the variable represents and whether negative values are physically or contextually meaningful. 7. In summary, domain restrictions depend on the scenario, not just the math. Always interpret the variables and context before deciding on domain and range. Final answer: Negative values in the domain are allowed only if they make sense in the scenario, such as position relative to a reference point like a cliff edge. Otherwise, restrict the domain to values that fit the real-world context, like $t \geq 0$ for time.