91Ó°ÊÓ

Multivariable functions are functions that depend on two or more variables. The functions discussed in this problem are multivariable, as they depend on variables like \( u, v, x, \) and \( y \). Each function takes inputs from multiple dimensions to output a single value or another function. Understanding how these functions interact with each other through composition is crucial. In our case, we first defined the function \( f(u, v) = uv - 3u + v \), where \( u \) and \( v \) act as inputs. We also defined two other functions \( g(x, y) = x - 2y \) and \( k(x, y) = 2x + y \) which depend on \( x \) and \( y \). The task is to see how the combination of these two results when they're plugged into \( f(u, v) \). Here is a brief look at the role of each function:

• **\( g(x, y) \)** transforms inputs \( x \) and \( y \) through subtraction and multiplication.
• **\( k(x, y) \)** transforms \( x \) and \( y \) through addition and multiplication.
• **\( f(u, v) \)** evaluates the results by combining them algebraically.

Algebraic manipulation is all about altering or rewriting expressions while keeping their values intact. In this problem, we need to manipulate the expressions to replace \( u \) and \( v \) with the values obtained from \( g(x, y) \) and \( k(x, y) \). We do this by:

• Substituting \( u = x - 2y \) and \( v = 2x + y \) into \( f(u, v) \).
• Using the distributive property and other algebraic laws to rewrite the resulting expression.

The expression becomes a bit complex, which is why step-by-step algebraic manipulation is essential. For example, the product \((x - 2y)(2x + y)\) is expanded by distributing each term. The expression is then cleaned up by removing parentheses, distributing terms, and combining like terms.By focusing on one operation at a time, you can significantly simplify complex expressions without errors. This process is essential in ensuring the correct final output of the function composition.

Simplifying expressions is the process of making them easier to understand or work with. This involves condensing them into their simplest form by combining like terms and reducing complexity where possible. The goal is to reach the simplest possible version of the expression while maintaining accuracy.In our problem, after performing substitutions and algebraic manipulations, the resulting expression still looks complex: \[ f(g(x, y), k(x, y)) = 2x^2 - 4xy + xy - 2y^2 - 3x + 6y + 2x + y \]To simplify it:

• Combine like terms, such as \(-4xy + xy\), resulting in \(-3xy\).
• Aggregate terms with variables like \(x\) and \(y\) separately (e.g., \(-3x + 2x = -x\) and \(6y + y = 7y\)).

Once fully simplified, the expression becomes:\[ 2x^2 - 3xy - 2y^2 - x + 7y \]This process ensures that the expression is efficient for further calculations or for analyzing its behavior.