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A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, a function is often represented as \( f(x) \), where \( x \) is an input variable. Functions are used to describe a real-world relationship in which one quantity depends on another.Functions can be visualized as machines that process an input to produce an output. You place a value into the function and receive a new value: the output. Functions have different applications in calculus, such as evaluating limits, finding derivatives, and integrating expressions. Here, in our original exercise, the function \( f(x, y) = 4x^{2}-xy-2y \) is a two-variable function. This means it depends on two inputs, \( x \) and \( y \), to calculate the output.

Polynomial expressions are mathematical expressions involving sums of powers in one or more variables multiplied by coefficients. For instance, \( 4x^2 - xy - 2y \) is a polynomial expression composed of three terms: \( 4x^2 \), \( xy \), and \( -2y \). Polynomial expressions can have different degrees, determined by the highest power of the variable in the expression.Let's break down what makes up a polynomial:

• A term like \( 4x^2 \) is composed of a coefficient (4) and a variable raised to a power (\( x^2 \)).
• Polynomial terms are additive, meaning they can be summed to build up the expression.
• Polynomials can have one or multiple variables. In our context, both \( x \) and \( y \) appear in the function's polynomial expression.

In calculus, polynomial expressions are crucial because they frequently arise in different calculus operations like differentiation and integration, and you often work with their roots and other properties.

Evaluating a function means finding the value of the function for specific input values. In our given exercise, the process involves two different evaluations. Each evaluation substitutes a different set of values into the function to find two outputs, \( f(x, x^2) \) and \( f(x, 1) \).To evaluate a function:

• Substitute the given values into the function formula.
• Perform the necessary arithmetic operations.
• Simplify the resulting expression to find the function's value.

In our problem:

• When evaluating \( f(x, x^2) \), we replaced \( y \) with \( x^2 \) in the function formula: \( f(x, x^2) = 4x^2 - x(x^2) - 2x^2 \). Simplifying gives us \( 2x^2 - x^3 \).
• For \( f(x, 1) \), replace \( y \) with 1, leading to \( f(x, 1) = 4x^2 - x(1) - 2 \). Simplifying that expression results in \( 4x^2 - x - 2 \).

Finally, the results of these evaluations are subtracted to give the desired outcome.Evaluating functions, especially polynomial functions, is a fundamental skill in calculus, as it helps in analyzing changes within a system or understanding the behavior of equations at specific points.