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When dealing with fractions, it's fundamental to understand what the denominator represents. In a fraction, the number on top is called the numerator, and the number on the bottom is the denominator. The denominator of a fraction tells us into how many equal parts the whole is divided.

For instance, in the function from our exercise, \(F(x) = \frac{x}{x - 1}\), the denominator is \(x - 1\). This part of the function is crucial because if the denominator ever equals zero, the fraction and the function as a whole become undefined. That's why identifying the denominator is always the first step in finding the domain of a function involving fractions.

An undefined function occurs when the function cannot produce a valid output for certain inputs. In the context of fractions, this typically happens when the denominator is zero, as you cannot divide by zero in the realm of arithmetic.

For our example function \(F(x) = \frac{x}{x - 1}\), we discover that when \(x\) is 1, the denominator becomes zero. That makes the function undefined at \(x = 1\), and we exclude this value from the domain. Remember, the absence of a valid denominator value results in an undefined function at that point. Undefined functions play a pivotal role in determining the limits and restrictions of a function's domain.

Interval notation is a mathematical shorthand used to describe sets of numbers along a number line. It is often used to specify the domain or range of a function. The notation includes two kinds of parentheses: round \(()\) and square \([]\). Round parentheses indicate that the endpoint is not included in the set, while square brackets mean the endpoint is included.

In our function's domain, \(F(x) = \frac{x}{x - 1}\), where \(x\) cannot be 1, we use round parentheses to show that 1 is excluded from the domain. The domain is then written as \((-\text{\infty} , 1) \cup (1, +\text{\infty})\), meaning all real numbers except 1 are included. Learning to read and write interval notation is indispensable for correctly communicating the domain of functions.