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When we talk about function operations in mathematics, we are referring to the various ways we can combine functions to create new functions. These operations include addition, subtraction, multiplication, and division. Just like with numbers, we can perform these operations on functions to obtain a resultant function.
For example, if we have two functions, say f(x) and g(x), we could create a new function by adding them together, f(x) + g(x), which would give us the sum of the two functions at any point x. This new function is called the sum of f and g, and we can plot it, differentiate it, integrate it, just as we could with the original functions.
In the context of the given problem, we are focusing on the multiplication (product) and division of functions, which allows us to create more complex functions and explore their relationships in deeper ways.

The function product is as straightforward as it sounds: it's the product of two functions. If we are given two functions, f(x) and g(x), their product is a new function h(x) defined by h(x) = f(x) \times g(x).
When we multiply these two functions, we perform the multiplication operation at each value of x, just as if we were multiplying two numbers. For instance, with f(x) = x - 1 and g(x) = \(\sqrt{x +1}\), the product function would be fg(x) = (x - 1) \(\sqrt{x +1}\).
This operation requires the domain of the new function to be the intersection of the domains of f(x) and g(x), since we need to ensure that the product is defined for all x values we are considering.

On the other hand, function division involves dividing one function by another, creating a function that represents the ratio of the two. If f(x) is our numerator function and g(x) is our denominator function, the division of f by g yields a new function h(x) defined by h(x) = f(x) / g(x), with the stipulation that g(x) cannot be equal to zero since division by zero is undefined.
In our textbook example, we are taking the function resulting from the product of f(x) and g(x) and dividing it by h(x). Thus, the function (fg)/h is expressed as (fg)/h(x) = \(\frac{(x - 1)\sqrt{x +1}}{2x^3 -1}\). It's important to note that when we divide functions, we are especially interested in the domain of the denominator function, as it dictates where the resultant function will be undefined due to division by zero.

The term algebra of functions refers to the set of rules and procedures that govern the combination of functions through operations like addition, subtraction, multiplication, and division. These rules are an extension of the algebra we perform with real numbers and are fundamental when working with functions in various fields of mathematics.
This algebra allows us to manipulate functions to form new functions and find relationships between them. It also comes in handy when solving complex equations and problems in calculus, like finding the derivative or integral of combined functions. In essence, the algebra of functions provides a framework for understanding and modeling relationships within mathematics that involve complex interaction between different functions, much like the ones we encounter in real-world problems.