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Understanding algebraic functions is a fundamental block in mastering algebra. An algebraic function is essentially a mathematical expression that relates an input, typically denoted as 'x', to an output, expressed as 'f(x)'. It's like a rule or a machine: you put an 'x' in, and the function provides an 'f(x)' out, according to some fixed, well-defined rule.

Take for example the functions mentioned in the exercise, where we have a linear function defined as \(f(x)=4x+2\), and another as \(g(x)=2x-9\). These are both simple examples of algebraic functions where the rule is a straightforward, linear relationship between the input and output. Algebraic functions can be more complex, involving quadratic terms (like \(x^2\)), cubic terms (like \(x^3\)), and so on. They can also include roots, exponential terms, and much more, but always following a specific rule tied to the 'x' input.

Now, function operations refer to the ways in which we can combine these rules. Think of it as having multiple machines, each with its own purpose, and you decide to make them work together. For algebraic functions, these operations commonly include addition, subtraction, multiplication, and division.

For our specific exercise, we are looking at the sum of two functions, which is a type of function operation. It involves adding the outputs of our function 'machines' for the same input 'x'. In mathematical terms, we see this as \((f+g)(x) = f(x) + g(x)\). It's critical to align the 'x' values, making sure we're referring to the same input point for both functions when we perform the addition. The fundamental principle here is that only the terms with the same degree can be added or subtracted. This is why we directly add \(4x\) from \(f(x)\) to \(2x\) from \(g(x)\), and the same for the constant terms \(2\) and \(-9\).

Polynomial arithmetic is a cornerstone of algebra, involving the addition, subtraction, multiplication, and occasionally division of polynomials. A polynomial is an expression made up of variables (like 'x') raised to various powers, and usually involves several terms added together.

By practicing polynomial arithmetic, students learn to work with algebraic functions that are polynomials—just like the linear functions in the exercise. Here, the process of adding them together is straightforward because they are both first-degree polynomials, also known as linear polynomials. The sum of \(4x+2\) and \(2x-9\) results in another polynomial, \(6x-7\), after combining like terms—those terms which have the same variable raised to the same power. Understanding how these terms combine is essential for working with functions and helps in recognizing patterns and simplifying complex expressions in algebra.