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Rational functions are a type of function that are expressed as the ratio of two polynomials. They take the general form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). The beauty of rational functions is that they can model a wide range of real-world phenomena, from physics to economics.

One important characteristic of rational functions is their domain. Since you cannot divide by zero, the domain of a rational function excludes any number that would make the denominator zero. This leads us to check which values might cause the function to become undefined.

To understand rational functions completely:

• Identify the polynomials in the numerator and the denominator.
• Find the zeros of the denominator by setting it equal to zero and solving for \( x \).
• Exclude these values from the domain.
  

This understanding assists in sketching graphs and analyzing how the function behaves across its domain.

In mathematics, an undefined point occurs when a function cannot produce a finite result for certain input values. In rational functions, this typically happens when the denominator equals zero, since division by zero is undefined.

To identify undefined points within a given rational function, follow these steps:

• Take the denominator and set it equal to zero, \( Q(x) = 0 \).
• Solve this equation for \( x \).
• The solutions will be the points where the function is undefined.
  

For the function \( f(x) = \frac{x-2}{x-2} \), the denominator \( x-2 = 0 \) when \( x = 2 \). This means the function is undefined at this point. Identifying undefined points is crucial for understanding and correctly interpreting the behavior of rational functions, ensuring accuracy in both mathematical exercises and practical applications.

Interval notation is a method of representing the domain (or range) of a function in a concise way. It uses intervals to show which parts of the real number line are included or excluded in a function's domain. An interval is written with brackets [ ], if the endpoint is included, or parentheses ( ), if it is not.

When expressing the domain of a function, you simply list the intervals that are included:

• Open interval: \((a, b)\) includes all numbers between \(a\) and \(b\), but not \(a\) and \(b\).
• Closed interval: \([a, b]\) includes all numbers between and including \(a\) and \(b\).
• Mixed interval: Combine open and closed, such as \([a, b)\) or \((a, b]\).
• Unbounded interval: Could extend to positive infinity \((a, \, \infty)\) or negative infinity \((-\infty, \, b)\), where infinity is always paired with a parenthesis.
  

For \( f(x) = \frac{x-2}{x-2} \), since \( x eq 2 \), the domain in interval notation is \( (-\infty, 2) \cup (2, \infty) \). This clearly indicates all real numbers except 2, highlighting the strength of interval notation in communicating an interval's precise boundaries.