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In mathematics, evaluating functions is a fundamental skill that enables us to find the output of a function for specific input values. This process involves substituting the given values into the function's algebraic expression and performing the necessary arithmetic operations. For example, consider the function \( z = f(x, y) = -11x + y \). To evaluate this function at the point \( (2,3) \), follow these steps:

• Identify the input variables \( x \) and \( y \).
• Substitute \( x = 2 \) and \( y = 3 \) into the function: \( z = -11(2) + 3 \).
• Perform the arithmetic: \( z = -22 + 3 = -19 \).

Therefore, \( f(2,3) = -19 \). The same steps are followed for other points, such as \( (11,1) \), demonstrating the systematic process of evaluating functions at given inputs.

Two-variable functions depend on two different input values, often denoted as \( x \) and \( y \). These functions are essential in various fields, including physics, economics, and engineering, as they can model situations where the output depends on multiple inputs. Consider the function \( z = f(x, y) = -11x + y \). This function takes two inputs, \( x \) and \( y \), and uses them to calculate an output \( z \).

• The structure \( f(x, y) \) highlights the dependence on both variables.
• The function combines the two inputs with specific coefficients, like \(-11\) for \( x \), effectively showing how each variable impacts the result.
• By evaluating the function at various points, we can understand how changes in \( x \) or \( y \) affect the outcome \( z \).

Two-variable functions are instrumental in solving real-world problems that involve multiple factors.

Algebraic expressions are combinations of variables, numbers, and arithmetic operations (such as addition, subtraction, multiplication, and division) that represent mathematical relationships. In the function \( z = f(x, y) = -11x + y \), the expression \(-11x + y\) is composed of the following:

• Variables \( x \) and \( y \), which represent unknown values or varying quantities.
• A constant \(-11\), which acts as a coefficient multiplying \( x \) to indicate its weight in the expression.
• An addition operation \(+\) that combines terms to calculate a final output.

Algebraic expressions serve as the foundational building blocks for defining mathematical functions. They illustrate how elements like variables and constants interact within an equation. When learning to interpret and manipulate algebraic expressions, it's important to follow the order of operations and understand how each part of the expression contributes to the whole. This understanding is crucial for successfully evaluating functions and solving complex mathematical problems.