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A rational function is defined as the quotient of two polynomials. In simpler terms, it's a fraction where the numerator and the denominator are both polynomials. For example, the function \( \frac{x-1}{x^{2}-5x+6} \) is a rational function. Here, \( x-1 \) is the numerator and \( x^2-5x+6 \) is the denominator.

One key aspect of rational functions is that they are not defined everywhere in the set of real numbers. The function becomes undefined at any value of \( x \) that makes the denominator zero. This is an important part of determining the function's domain.

To find where a rational function is undefined, you need to solve the equation where the denominator equals zero. This process identifies the values of \( x \) that are not in the domain. Once these values are known, they can be excluded from the set of real numbers to form the domain of the function. Understanding these exclusions is crucial whenever you're working with rational functions.

Quadratic equations are polynomial equations of the second degree. They are usually in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The solutions to these equations are known as the 'roots'.

To solve a quadratic equation, one common method is factoring. This involves rewriting the quadratic in a form where it can be split into two binomials. In the example \( x^2 - 5x + 6 \), it factors into \( (x-2)(x-3) \).

• Setting each factor equal to zero, \( x-2 = 0 \) and \( x-3 = 0 \), provides the solutions or roots.
• Solving these gives the values \( x = 2 \) and \( x = 3 \).

These solutions are particularly significant when they appear in the denominator of a rational function because they mark the points where the function is undefined. Therefore, including or excluding these solutions from the domain in subsequent steps is vital for accurately determining the domain.

Interval notation is a mathematical notation used to represent subsets of real numbers. It provides a clear visual format to express the domain or range of a function.

In interval notation, brackets and parentheses are used to denote the inclusion or exclusion of endpoints.

• '(' or ')' means that the endpoint is not included, also known as open interval.
• '[' or ']' indicates that the endpoint is included, known as closed interval.

In the exercise example, the function \( \frac{x-1}{x^{2}-5x+6} \) is undefined at \( x = 2 \) and \( x = 3 \). Thus, the domain excludes these points and is written in interval notation as \( (-\infty, 2) \cup (2, 3) \cup (3, \infty) \).

Using interval notation helps to succinctly communicate which values are included in the domain of a function, providing an efficient way to express complex restrictions on input values.