Let the function \(f(a, b)\) be defined as \(f(a, b)=b^{2}-a\) For all \(x\) and \(y, f\left(\left(x^{2}+y^{2}\right),(x-y)\right)=?\) F. 2\(y^{2}\) G. 0 H. \(-2 y^{2}\) J. \(-2 x y+2 y^{2}\) K. \(-2 x y\)

In mathematics, a function is a relationship between two sets that assigns to every element of the first set exactly one element of the second set. Think of it as a machine that takes an input, does something to it, and then produces an output. The function we are discussing in this ACT Math practice question is written as f(a, b) = b² - a. The inputs for this function are 'a' and 'b', and it produces an output by squaring 'b' and subtracting 'a'.

When we apply this function to the input pair \( (x^2 + y^2), (x-y) \), we are essentially feeding this machine a snack made of other mathematics operations - and we need to see what comes out the other side. The function remains the same; it's the inputs that are getting more complicated. In order to process these inputs, we need to perform a series of algebraic manipulations, which leads to the concept of simplification.

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the given problem, we encounter algebraic expressions when we input the expressions \(x^2 + y^2\) for 'a' and \(x-y\) for 'b' into the function.

To effectively work with these expressions during the ACT Math practice, there are a few key strategies to keep in mind:

• Substitution: As we did in step 1 of the solution, we replace 'a' and 'b' in the function with the given expressions.
• Expansion: When we encounter a squared term, such as \( (x-y)^2 \), we expand it to \( x^2 - 2xy + y^2\) (step 2 of the solution). This helps us to see and combine like terms.
• Combining like terms: After expanding, we combine the terms that are alike (such as \(x^2\) terms and \(y^2\) terms) to simplify the expression further (step 3 of the solution).

Understanding how to manipulate algebraic expressions is crucial for simplifying them to find the correct answer.

The concept of simplification in algebra involves reducing an expression to its most basic form. Simplification can involve combining like terms, factoring, expanding expressions, and canceling out terms. In our exercise, simplification comes into play after we have substituted our variables and expanded the square term. We are left with an expression that looks complex, but upon careful inspection, we notice that terms can actually be subtracted from each other to leave us with a more concise result.

In step 3 of the solution, the simplification process is where we subtract \(x^2 + y^2\) from \(x^2 - 2xy + y^2\) to ultimately find that our function simplifies to \( -2xy \). This final simplified form allows us to correctly identify the solution (K) from the list of options provided. Simplification is not only about making an expression shorter; it's about stripping it down in such a way that the true essence of what the expression represents is easier to comprehend and utilize.