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In mathematics, a function is a special kind of relation that assigns each input exactly one output. Functions are usually written as \( f(x) \), where \( x \) is the input, and \( f(x) \) is the output. For example, in the problem given, we have the function \( f(a) = a^2 - 12 \). Here, \( a \) represents the input, and \( a^2 - 12 \) is what comes out after applying the function to \( a \).
Expressions, on the other hand, are combinations of numbers, variables, and arithmetic operations. They do not have an equal sign. In the function \( f(a) = a^2 - 12 \), the expression part is \( a^2 - 12 \). This tells us what calculations are performed using the input \( a \).
When dealing with functions and expressions, it helps to understand how to substitute values and simplify them:

• Substitute given values into the function in place of the variable.
• Manipulate expressions according to algebraic rules to simplify or find specific values.

Understanding these concepts forms the foundation for tackling more complex math problems involving functions.

Binomial expansion is a method used to expand an expression that raises a binomial to a power. A binomial is an algebraic expression containing two distinct terms. In the exercise, the expression \((b-a)^2\) is an example of a binomial where binomial expansion is applied.
When expanding \((b-a)^2\), you apply the formula \((b-a)^2 = b^2 - 2ab + a^2\). Each term in the expansion represents how parts of the binomial multiply together.

• \(b^2\) comes from multiplying \(b\) by itself.
• \(-2ab\) comes from two instances of \(b \times -a\).
• \(a^2\) comes from multiplying \(-a\) by itself.

Expanding binomials simplifies expressions, making them easier to integrate into other functions or equations. The understanding of how to expand a binomial is vital because it allows you to break down expressions into simpler components that are easier to analyze and solve.

Algebraic manipulation involves rearranging expressions using basic algebraic rules to solve equations or simplify expressions. This is a crucial skill in solving math problems.
In the problem provided, after expanding the binomial, we need to substitute the expanded expression back into the function. The process involves:

• Substituting \((b-a)^2\) with its expanded form \(b^2 - 2ab + a^2\).
• Integrating the new expression into the function \(f(b-a) = (b-a)^2 - 12\).
• Simplifying to get \(a^2 - 2ab + b^2 - 12\), which matches the correct answer choice.

By mastering algebraic manipulation, you can effectively handle expressions and equations, making it much easier to find solutions to complex problems. Practice involves recognizing which operations to apply and executing them step by step while maintaining mathematical accuracy.