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Polynomial functions are at the heart of algebra. They are mathematical expressions that involve sums of powers of variables with coefficients. The general form of a polynomial function is:

• Constant term: a number by itself. For example, "+2".
• Linear term: a variable without an exponent, like "x".
• Quadratic term: a variable with an exponent of 2, such as "x^2".

In the context of function operations, we often look at polynomial functions such as:

• \( f(x) = x - 6 \)
• \( g(x) = 2x^2 \)

Here, \( f(x) \) is a linear polynomial because it primarily involves the first power of x, while \( g(x) \) is quadratic due to the presence of the \( x^2 \) term. When we perform operations on functions like these, we combine or manipulate their terms systematically. This leads to new polynomial expressions like \( (f+g)(x) = 2x^2 + x - 6 \). These operations can show how two simpler polynomial functions combine to form a more complex one.

Function notation is a way to represent mathematical equations in terms of functions. It's a handy tool for expressing relationships and calculations without using a lot of words. At its core, function notation is simple:

• The notation \( f(x) \) indicates a function named 'f' in terms of the variable 'x'.
• Similarly, \( g(x) \) represents a function named 'g'.

When you see expressions like \( (f+g)(x) \), this tells you to perform an operation (in this case, addition) on the functions \( f \) and \( g \). In our exercise, this leads to

• Adding the outputs of \( f(x) = x - 6 \) and \( g(x) = 2x^2 \).

This results in a new function: \( (f+g)(x) = 2x^2 + x - 6 \). Function notation provides clarity when dealing with multiple operations and functions, making it easier to interpret and analyze mathematical expressions.

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They're the building blocks of many mathematical concepts. When dealing with algebraic expressions, consider the following components:

• Variables represent unknowns or values that can change, like "x".
• Coefficients are numbers multiplied by these variables, such as the "2" in "2x^2".
• Operators like +, -, *, and / show how the numbers and variables are to be combined or separated.

In our problem, \( f(x) = x - 6 \) and \( g(x) = 2x^2 \) are algebraic expressions. The task of finding \((f+g)(x)\) involves adding these expressions together. It's crucial to remember that different types of terms, such as linear \((x)\) and quadratic \((2x^2)\), cannot be further simplified if they are unlike. Hence, when you combine \((x - 6)\) and \(2x^2\), you simply write them out as \(2x^2 + x - 6\). The result is a well-formed algebraic expression that is ready for analysis or further operations if needed.