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A rational function is a type of function that can be expressed as the quotient of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. For example, the function \( g(x) = \frac{2}{x-1} \) is a rational function. Here, the numerator is a constant polynomial \( 2 \), and the denominator \( x-1 \) is a linear polynomial.

The most interesting feature of rational functions is that they have vertical asymptotes. A vertical asymptote occurs when the denominator equals zero, causing the function to be undefined at that point. In our example, setting \( x-1 = 0 \) reveals \( x = 1 \) as the point where the function is undefined. This means that the graph of the function will never touch or cross the vertical line \( x = 1 \).

Rational functions also have specific behaviors at infinity, often having horizontal or oblique asymptotes as \( x \) approaches positive or negative infinity. These aspects are crucial for understanding the domain and range of these functions.

Function notation is a way to denote functions in mathematics using symbols for easier understanding and manipulation. Typically, we use the letter \( f \), \( g \), or \( h \) to name a function. For instance, in our example, \( g(x) = \frac{2}{x-1} \), \( g \) represents the function and \( x \) represents the input or the variable.

In function notation, \( g(x) \) is read as \"\( g \) of \( x \)\" and represents the output of the function when \( x \) is the input. This notation is very important because it allows us to specify which variable is considered the input and which expression represents the output easily.

Using function notation helps in defining and analyzing various properties of functions, such as their domain and range. It provides a clear and concise method to indicate which variable is being manipulated.

Real numbers are a fundamental concept in mathematics, covering all the numbers you can think of on the continuous number line. They include all the rational numbers like integers \( -3, 0, 1 \), fractions \( \frac{1}{2}, \frac{2}{3} \), and the irrational numbers such as \( \sqrt{2} \) and \( \pi \).

In the context of rational functions, understanding real numbers is essential for defining the domain and range. For instance, the domain of \( g(x) = \frac{2}{x-1} \) is all real numbers except where the denominator becomes zero. This is typically the case with rational functions as they tend to be undefined at certain points. Therefore, the domain is expressed as \( (-\infty, 1) \cup (1, \infty) \), indicating all real numbers except \( x = 1 \).

As for the range, since the values \( g(x) \) can take after substitution are all possible real numbers except where division by zero would occur or where it is undefined, it often spans \( (-\infty, \infty) \). This characteristic of including all permissible real numbers is what makes real numbers so critical in understanding and quantifying the behavior of functions.