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Evaluating a function involves finding the output value for a given input value. This is a crucial concept in understanding how functions work. When we evaluate a function at a specific point, we determine the corresponding result according to the rules set by the function equation. In this case, our function is given as:\[ h(t) = -t^3 - 2t^2 + 3 \]To evaluate, plug in different values of \( t \) to find specific outputs. This gives insight into the behavior of the function across different inputs. Function evaluation is essential in mathematics as it provides concrete examples of how a function operates.

Substitution is the action of replacing a variable in a function with a specific value. It is a straightforward process and fundamental in evaluating functions. In the problem we're discussing, substitution means replacing \( t \) with a number such as \(-3\), \(0\), or \(2\). For example, substituting \( t = -3 \) in the function \( h(t) = -t^3 - 2t^2 + 3 \) leads to:\[ h(-3) = -(-3)^3 - 2(-3)^2 + 3 \]Substitution transforms the abstract function form into a numerical form, thus allowing calculation and evaluation. This concept is closely tied to algebraic manipulation and is a foundational skill in both mathematics and sciences.

Simplification is the process of reducing an expression to its most basic form. After substitution, the expression is often a mix of terms and operations that must be simplified. This involves performing arithmetic operations like exponents and multiplication to reach a final single-number result.For \( h(-3) \), after substitution, the expression is:\[ -(-3)^3 - 2(-3)^2 + 3 \]Simplifying step by step, starting with:- \((-3)^3 = -27\) then negating: \(-(-27) = 27\)- \((-3)^2 = 9\) multiplied by \(-2\): \(-18\)So, the simplified result becomes:\[ 27 - 18 + 3 = 12 \]This method highlights the importance of following the order of operations in mathematics.

A cubic function is a type of polynomial function with the highest power of the variable as three, represented as \( ax^3 + bx^2 + cx + d \). It includes an \( x^3 \) term and is generally shaped like an 'S' curve in its graph, possibly having one or two turns, and can intersect the x-axis up to three times.The given function \( h(t) = -t^3 - 2t^2 + 3 \) is a cubic function because of the \(-t^3\) term. This type of function allows more complex behavior than linear or quadratic functions and can model a wider range of phenomena in real-world applications. A cubic function’s graph gives us visual insight into how quickly and in which direction the function's value changes as \( t \) changes.