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A rational function is a type of function that can be expressed as the ratio of two polynomials, where the numerator and the denominator are both polynomials. In mathematical forms, it looks like this: \( f(x) = \frac{p(x)}{q(x)} \) where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \) is not the zero polynomial. The simplest example is \( f(x) = \frac{1}{x} \) where the numerator is the constant 1 and the denominator is the polynomial \( x \).

The characteristics of a rational function, such as its domain, asymptotes, and intercepts, are often explored in algebra and precalculus courses. One crucial aspect in these characteristics is understanding the domain of the rational function, or the set of all possible values of \( x \) that will produce a real number output when substituted into the function. A clear understanding of the domain helps avoid undefined expressions and sets the foundation for graphing the function accurately.

The concept of excluded values is essential when discussing the domain of rational functions. Excluded values are specific numbers that make the denominator of a rational function equal to zero. Since division by zero is undefined in mathematics, such values must be excluded from the domain of the function.

In the example \( f(x) = \frac{1}{x+8} + \frac{3}{x-10} \) the excluded values are found by setting each denominator equal to zero and solving for \( x \). This process yields two critical numbers: \( x = -8 \) and \( x = 10 \). These values cannot be part of the domain because they make the function undefined. Remember that identifying these excluded values is a key step in finding the correct domain of a rational function.

Interval notation is a system of writing the domain or range of a function that involves using intervals to describe a continuum of numbers. It's an efficient way to represent bounds and is commonly used in calculus and real analysis. The main symbols in interval notation include parentheses '()' for excluded boundaries, and brackets '[]' for included boundaries.

For the rational function example above, the domain excludes the numbers -8 and 10, so the interval notation would be \( (-\infty, -8) \cup (-8, 10) \cup (10, \infty) \). This notation tells us that the domain includes all real numbers except for -8 and 10, and the union symbol '∪' indicates that the domain is comprised of multiple separate intervals. Understanding this notation is crucial as it provides a clear and concise way to communicate the set of inputs for which the function is defined.