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In mathematics, a two-variable function is an expression where two variables serve as inputs. These variables usually have designated names like \(x\) and \(y\), and the function calculates a single output based on specific values for these inputs. Two-variable functions are often represented in the form \(f(x, y)\).

In our example, the function is \(f(x, y) = 3x - 4y\). This tells us that for each pair of values, \((x, y)\), there is a unique output generated by applying the formula given. The numbers before \(x\) and \(y\) in the expression are constants, which indicate how each variable contributes to the function's output.

• Constant 3 multiplies by \(x\): This means the effect of \(x\) on the output is threefold.
• Constant -4 multiplies by \(y\): Similarly, the effect of \(y\) is fourfold, but also has a negative sign, indicating subtraction.
  

Understanding the specific relationship between the inputs \(x, y\) and the output helps make two-variable functions an essential tool in modeling real-world situations where two factors influence a dependent result.

Substitution in functions involves replacing the variables in the function's expression with specific given values. This process allows us to evaluate the function for particular inputs. In the provided exercise, we substitute \(x = -2\) and \(y = 3\) into the function \(f(x, y) = 3x - 4y\).

Here's how substitution works step by step:

• Identify each variable in the function's expression. Our function has two variables: \(x\) and \(y\).
• Replace \(x\) with the given value, \(-2\).
• Replace \(y\) with the given value, \(3\).
• The expression transforms into \(f(-2, 3) = 3(-2) - 4(3)\).

By following these steps, we tailor the function to reflect these particular values, paving the way to perform numerical calculations, which ultimately provide us with the function's output.

Algebraic expressions form the backbone of algebra and consist of variables, constants, and arithmetic operations. In contexts like evaluating functions, understanding how to manipulate these expressions is crucial.

For the function \(f(x, y) = 3x - 4y\), we focus on applying arithmetic operations like multiplication and subtraction to find the output. Here are the detailed steps we follow for our exercise:

• Multiplication: Calculating the sections \(3(-2)\) and \(-4(3)\).
• The product of \(3(-2)\) gives \(-6\), and \(-4(3)\) gives \(-12\).
• Subtraction: We then subtract these results to combine them into a single final value: \(-6 - 12\).
• This operation leads us to the result: \(-18\).

Understanding these basic arithmetic operations on algebraic expressions is key to evaluating any function effectively, and provides a reliable foundation for more complex mathematical problems.