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Polynomial functions are foundational to algebra and appear in many different areas of mathematics and real-world applications. A polynomial function is an expression constructed from variables and constants, employing only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

For example, the given functions in our exercise, f(x) = x^3 + 5 and g(x) = x^2 - 2, are both polynomial functions. The highest exponent of the variable x determines the degree of the polynomial; hence f(x) is a third-degree polynomial, and g(x) is a second-degree polynomial.

It's important for students to recognize the structure of polynomial functions because they obey certain rules that make operations like addition, subtraction, and multiplication predictable and structured, facilitating easier manipulation and understanding of these expressions.

Function subtraction is an operation where you find the difference between corresponding terms of two functions. In the context of polynomial functions, like the ones provided in our exercise, the operation is carried out by subtracting the polynomial expressions for each respective value of x.

First, write out the functions explicitly. For our example, (f - g)(x) means to take the function f(x) and subtract the function g(x) from it. The subtraction is to be done term-by-term, and it's crucial to pay attention to the signs when performing the subtraction. Remember to properly subtract any constants and to align like terms before combining them to simplify the expression.

Algebraic manipulation involves the strategic use of mathematical operations and properties to simplify or rearrange algebraic expressions. It's an essential skill for solving algebra problems, like function operations we are dealing with.

When subtracting functions, as in our exercise, distributing the negative sign across the terms is a key manipulation step that helps simplify the resultant function. Always watch for terms that can be combined (like terms) and those that cannot. For example, after properly distributing the negative sign in (f-g)(x), we combine the like terms to arrive at the simplified function. Mastery of algebraic manipulation is not just about knowing the rules, but also about recognizing patterns that allow for simplification of complex algebraic expressions.