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Rational functions are a special category of functions that students often encounter in algebra. A rational function is any function that can be written as a fraction where both the numerator and the denominator are polynomials. Keep in mind:

• The numerator is the top part of the fraction.
• The denominator is the bottom part of the fraction.

For example, in the function \(h(x) = \frac{x+8}{x^2-64}\), the numerator is \(x + 8\) and the denominator is \(x^2 - 64\).
Rational functions are defined wherever the denominator is not zero. Identifying where a function is undefined is crucial in determining its domain. Hence, understanding how to work with rational functions prepares students for finding where the function has undefined points and, eventually, its domain.

In the context of rational functions, undefined points are values of \(x\) that make the denominator equal to zero. When the denominator becomes zero, the function cannot be evaluated as it leads to division by zero, which is undefined in mathematics.
Let's illustrate this with the function \(h(x) = \frac{x+8}{x^2-64}\). To find the undefined points, set the denominator equal to zero: \(x^2 - 64 = 0\). Solving this equation, we get \(x^2 = 64\). Taking the square root gives two solutions: \(x = 8\) and \(x = -8\). Thus, \(h(x)\) is undefined at \(x = 8\) and \(x = -8\).

• Whenever you solve for undefined points, make sure to exclude these values when determining the domain of the function.

Recognizing these points helps to clearly state where the function cannot be evaluated, which is an essential step in finding the domain.

Interval notation is a method used in mathematics to describe the set of numbers that make up the domain of a function. It is a concise way to express which values are included or excluded from the set.
When writing interval notation, square brackets \([, ]\) are used to denote that an endpoint is included, whereas parentheses \((, )\) indicate that an endpoint is not included.
For the function \(h(x) = \frac{x+8}{x^2-64}\), we've already identified that \(x = 8\) and \(x = -8\) are undefined points. Therefore, the domain is expressed in interval notation as:

• \((-\infty,-8)\): All numbers less than -8.
• \((-8,8)\): All numbers between -8 and 8.
• \((8,\infty)\): All numbers greater than 8.

Each part is combined using the union symbol \(\cup\), reflecting the entire set of allowable values for \(x\), minus the excluded points. Mastering interval notation allows students to represent domains clearly and accurately.