91Ó°ÊÓ

In mathematics, the domain of a function is the complete set of possible values of the independent variable. In simpler words, it's everything you can put into a function that the function can handle and compute results for. For example, if we have a function like

• \( h(x) = \sqrt{x} \)

we can only use non-negative numbers as inputs, since the square root of a negative number is not defined in the real numbers.
Thus, the domain here would be all non-negative real numbers.

When working with polynomial functions, things are a bit simpler. Polynomials are defined for all real numbers; they have no restrictions.

However, with a division of two functions, like with the expression \( \frac{f(x)}{g(x)} \), you need to make sure the denominator \( g(x) \) is not zero.
Wherever the denominator is zero, we have to exclude this from the domain.

Polynomial functions are a type of mathematical function that include terms like powers of \(x\). A term in a polynomial is the product of a number, called the coefficient, and a variable raised to a whole number power. For example,

• \( f(x) = x^3 - 4x + 7 \)

is a polynomial function with three terms.
The powers in polynomial functions are always non-negative integers. They can also be summed together with other similar terms.

These types of functions are very versatile and can model a wide range of situations in mathematics and real life.

Importantly, for our exercise, polynomials are defined everywhere in the real number system. They don’t have any quirky rules or limitations on the values that \(x\) can take. Thus, the domain is all real numbers for any polynomial function.
This makes them simpler to work with compared to other kinds of functions, such as rational or radical ones.

Function division occurs when you divide one function by another. In mathematical notation, if \(f(x)\) and \(g(x)\) are two functions, then their division is represented as \( \frac{f(x)}{g(x)} \). It involves both the numerator and denominator functions. For example, if

• \( f(x) = x - 3 \)
• \( g(x) = x^2 \)

then the division would look like, \( \frac{x - 3}{x^2} \).

One crucial rule for this operation is ensuring that the denominator \( g(x) \) does not equal zero.
If it does, the function becomes undefined at those points, and we must exclude them from the domain.

For the given function division, \( g(x) = x^2 \) becomes zero at \( x = 0 \), so we exclude zero from the domain. The result is a domain of all real numbers except zero, denoted as \( \mathbb{R} \setminus \{0\} \).