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The domain of a function is the set of all possible input values (usually represented by "x") that will result in a valid output for the function. For polynomial functions, the domain is typically all real numbers because polynomials are defined for every real x-value. This kind of function has no restrictions like square roots or denominators that could be zero.
For example, the functions given in the exercise, \( f(x) = x^2 + 2x \) and \( g(x) = 6 - x^2 \), are both polynomial functions.
\(f(x) \) can take any real number as its input, and so can \(g(x)\), making their domains \((-\infty, \infty)\).

• When combining functions, calculating the domain of an operation like \( f+g \) or \( f \cdot g \) is straightforward because it remains all real numbers \((-\infty, \infty)\).
• However, for a division operation like \( \frac{f}{g} \), the domain can be more restricted because division by zero is undefined.
• In this exercise, \( g(x) = 6 - x^2 \) becomes zero when \( x = \pm\sqrt{6} \). So, the domain of \( \frac{f}{g} \) excludes these x-values, resulting in the interval \((-\infty, -\sqrt{6}) \cup (-\sqrt{6}, \sqrt{6}) \cup (\sqrt{6}, \infty)\).

Polynomial functions are expressions that consist of variables and coefficients to describe the sum of multiple terms, where each term is a product of a constant and a power of a variable. These functions are characterized by their general form:
\[ a_n x^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1 x + a_0 \]
where the \(a_n\) coefficients are real numbers and \(n\) is a non-negative integer. The degree of the polynomial is the highest power of the variable present.
In the given exercise:

• \(f(x) = x^2 + 2x\) is a polynomial of degree 2, with the highest power being \(x^2\).
• \(g(x) = 6 - x^2\) is also a polynomial of degree 2, but written differently, with the constant term appearing first.

Polynomials are some of the simplest functions to handle when combining because they are defined for all real x-values. This makes function operations with polynomials, such as addition, subtraction, and multiplication, straightforward in terms of domain determination.

Function operations involve combining two functions in various ways to produce a new function. This can include operations like addition, subtraction, multiplication, and division.

• **Addition and Subtraction**: These are the simplest operations. When you add or subtract two functions, you combine like terms. For example, adding \( f(x) = x^2 + 2x \) and \( g(x) = 6 - x^2 \) results in \( (x^2 + 2x) + (6 - x^2) = 2x + 6 \), a linear function.
• **Multiplication**: In multiplication, you distribute each term in the first function over every term in the second one. For \( f(x) \cdot g(x) \), multiple steps are required, like calculating \(x^2 \cdot 6\) and \(2x \cdot (-x^2)\), resulting in a degree 4 polynomial.
• **Division**: Division is more complex because it requires the denominator to never be zero. This affects the domain. For instance, \( \frac{f(x)}{g(x)} = \frac{x^2 + 2x}{6 - x^2} \) became undefined at \( x = \pm\sqrt{6} \).

Performing each operation accurately will not only combine the functions correctly but also identify any changes to their domains.