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Inverse functions are two functions that 'undo' each other. For any function to have an inverse, each output must be uniquely paired with an input—that is, the function must be one-to-one (bijective). When you compose a function with its inverse, the result should be the identity function, meaning you get the original input back: \(f(g(x)) = g(f(x)) = x\).

To verify if two functions are inverses, we perform two composition operations. First, we substitute the second function into the first and simplify to see if we get the identity function. Next, we repeat this process with the order of the functions reversed. If in both cases the identity function is obtained, then the functions are indeed inverses of each other. In the given exercise, both compositions \(f(g(x))\) and \(g(f(x))\) did not yield the identity function, demonstrating that \(f\) and \(g\) are not inverse functions of one another.

Function operations include operations such as function composition, addition, subtraction, multiplication, and division. Composition of functions is a type of operation where one function is applied to the result of another function. In mathematical terms, given two functions \(f(x)\) and \(g(x)\), their composition is denoted as \(f(g(x))\) or \(g(f(x))\), depending on the order of application.

When composing functions, it's essential to pay attention to the order since function composition is not necessarily commutative; that is, \(f(g(x))\) may not equal \(g(f(x))\). In solving the provided exercise, the step-by-step solution first composed the functions one way and then the other. Simplifying these composite functions helps determine if the original functions have the inverse relationship.

Precalculus algebra is a branch of mathematics that prepares students for the study of calculus. It covers various concepts including functions and their properties, inverse functions, polynomials, exponentials, and logarithms. Understanding how to manipulate functions and find their inverses is critical in precalculus.

Within precalculus, knowing how to accurately compose and decompose functions is a vital skill. When given an exercise like the one we have, one uses algebraic manipulation to simplify expressions after composing functions. By performing operations such as distributing, combining like terms, and cancelling, one simplifies the composite functions. The step-by-step solution applied these algebraic techniques to demonstrate that \(f(x)\) and \(g(x)\) were not inverses, emphasizing the precalculus skill of transforming and understanding the structure of functions.