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Understanding how to simplify functions is crucial when dealing with complex mathematical expressions, especially in algebra. In the context of our exercise, function simplification helps to transform the given function, which initially seems complicated, into a more manageable form.

Simplifying functions often involves combining like terms, factoring, expanding polynomials, or even canceling out common factors. The aim is to make the function as straightforward as possible, without changing its value. When simplifying the difference of two functions, like in our exercise \(g(x) - g(3)\), we subtract the values of these functions at different points, which requires us to carefully combine these values into a single expression. It's like taking two complex pieces of a puzzle and fitting them together to make a clearer picture. The simpler the function, the easier it is to work with for further analytical or computational operations.

The least common multiple (LCM) is a fundamental concept that refers to the smallest number that is a multiple of two or more numbers. To put it simply, it's the smallest 'shared' number that a set of numbers can all be multiplied into without leaving a remainder.

In algebra, finding the LCM of two or more denominators is a vital step when adding, subtracting, or comparing algebraic fractions. This step facilitates the process of combining fractions over a common denominator. For example, when the exercise asks us to find the LCM of \(x^2\) and 9, we aim to identify a number or, in this case, a polynomial expression, which both \(x^2\) and 9 can evenly divide into. The LCM can help turn a complex fraction into a more straightforward single fraction, as seen in the step-by-step solution.

Algebraic fractions are very similar to the fractions we encounter with whole numbers, but they involve variables like \(x\). Just like with numerical fractions, we can perform operations such as addition, subtraction, multiplication, and division with algebraic fractions.

In the given exercise, we deal with the division of two algebraic fractions. This operation is handled by multiplying the first fraction by the reciprocal of the second. The process requires a good grasp of fraction multiplication and division rules. Simplifying algebraic fractions often involves factorizing numerators and denominators and canceling out common factors to reduce the fraction to its simplest form. The goal is to make the expression cleaner and more understandable, which inevitably leads to a smoother problem-solving process.