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Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In high school algebra, symbols often represent numbers in equations and formulas. The ability to simplify, organize, and solve these equations is crucial for understanding a variety of mathematical concepts.

When working with problems involving function multiplication, algebra is used to combine and simplify expressions. This involves applying techniques like distributing and combining like terms. For instance, when calculating the product of two functions, each term of the first function is multiplied by every term of the second function. This results in a larger expression that then needs to be simplified.

In this exercise, we apply algebraic principles to multiply two linear functions, resulting in a quadratic expression. Understanding basic algebraic operations like these enables you to handle more complex mathematical scenarios.

The product of functions involves multiplying two functions together to arrive at a new function. If you have two functions, say, \( f(x) \) and \( g(x) \), the product is written as \( (f \cdot g)(x) = f(x) \cdot g(x) \).

The idea when multiplying these functions is to consider each function as a whole and multiply them using algebraic methods. In many cases, like here, both functions are polynomials. You multiply each part of one polynomial by each part of the other, following the distributive property of multiplication over addition.

For this particular problem, the functions \( f(x) = 2x - 3 \) and \( g(x) = 3x - 1 \) are multiplied using this method, resulting in the expression \((2x - 3)(3x - 1)\). Once expanded, this gives \(6x^2 - 2x - 9x + 3\), which simplifies to a more manageable form \(6x^2 - 11x + 3\).

Understanding function product provides a foundation for more advanced concepts, such as the integration of functions and dynamic systems modeling.

Function evaluation is the process of calculating the output of a function given a specific input. It's like plugging a number into a machine and watching it produce a result based on a predefined rule.

In mathematical terms, if you have a function \(f(x)\), evaluating it at \(x = c\) means substituting the number \(c\) into every instance of \(x\) in the function. After substitution, you perform the necessary arithmetic operations to find the function's value.

For this exercise, once we discover the equation for \( (f \cdot g)(x) \) as \(6x^2 - 11x + 3\), the next step is to evaluate it at \(x = 12\). You simply replace \(x\) with 12 in the expression and calculate: \(6(12)^2 - 11(12) + 3\). Solving this gives the value of \((f \cdot g)(12)\).

Function evaluation is a fundamental skill in algebra that allows you to explore how functions behave and is a step towards understanding calculus, where function analysis becomes more complex.