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Algebraic functions are the building blocks of mathematical expressions that involve algebraic operations. They are functions defined by polynomial equations or the combination of polynomials using various mathematical operations. In the context of this exercise, we are dealing with simple algebraic functions such as the linear function \(f(x) = x - 1\) and a more complex function like \(g(x) = \sqrt{x+1}\), which combines a square root operation. These functions can vary in complexity, but their foundational rules are set by the algebraic operations and structures they are built upon. The task of combining these functions teaches us how to handle algebraic expressions and functions in a coherent manner, forming a new function from them.

In mathematics, operations are procedures or actions that produce some form of output from given inputs. Common operations include addition, subtraction, multiplication, and division. In functions, performing these operations modifies the inputs to produce desired outcomes. For instance, in the exercise, we perform addition:

• Addition is used to combine the output of two functions \(f(x) = x-1\) and \(g(x) = \sqrt{x+1}\).

This is seen in \[(f+g)(x) = f(x) + g(x) = (x-1) + (\sqrt{x+1})\]
By adding these functions, we achieve a composite result that comprises elements of both parent functions. Understanding how to perform mathematical operations on functions is essential to creating new functional expressions and solutions.

Function rules provide the framework that describes how to process inputs into outputs within a given function. Each function has its own unique rule, which specifies the operations needed to transform an input \(x\) into an output \(f(x)\). For example:

• The rule of the function \(f(x) = x - 1\) signifies subtracting 1 from the input.
• The rule of the function \(g(x) = \sqrt{x+1}\) indicates taking the square root of one added to the input.

When combining functions, as in the exercise, the rules of each function are merged. The rule for \((f+g)(x)\) is a direct result of adding the rules:\[(f+g)(x) = (x-1) + (\sqrt{x+1})\]
This requires handling the combined operations seamlessly, while ensuring adherence to each function’s original rule.