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When we talk about function evaluation, we are essentially looking at the process of determining the output of a function for a particular input. This is a fundamental skill in mathematics as it allows us to understand how a function behaves across different values.

For the function given in the exercise, which is a quadratic function represented by the formula (f(x) = -x^2), we want to determine the output when we substitute specific values for (x). To make this concept clearer, imagine that (f(x)) is a machine that transforms numbers. When you insert (x = 1) into this machine, it squares the number and then multiplies it by (-1), giving us the output (-1).

This process would be repeated for each value of (x) provided in the exercise, which illustrates the essence of function evaluation: input a value, and using the function's formula, find out the output.

Quadratic functions are a special class of polynomial functions that can be recognized by the highest degree of the variable being two. The general form of a quadratic function is (f(x) = ax^2 + bx + c), where (a, b), and (c) are constants, and (a ≠ 0).

The function in our exercise is a simple quadratic function where the coefficients (b) and (c) are zero, leaving us with (f(x) = -x^2). Quadratic functions are known for their curved graphs called parabolas, and in the case of our exercise, the graph opens downwards since the coefficient of (x^2) is negative. This is why the output values are becoming more negative as the value of (x) increases.

The substitution method is a technique used to evaluate functions. It involves replacing the variable in the function's formula with a given number to calculate the output. This is exactly what we did in the step-by-step solution, where we substituted values (0, 1, 2, 3,) and (4) for (x) one by one into the function (f(x) = -x^2).

The key to successfully applying the substitution method is to ensure that every instance of the variable (x) is replaced by the given value and that all operations are followed according to the order of operations. In the provided exercise, the steps highlight this method clearly, showing that regardless of the function's complexity, substitution is a straightforward process of replacing and calculating.