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In rational functions like \( f(x) = \frac{1}{x-3} \), identifying the domain and range is a key step. The domain refers to all possible input values \( x \) for which the function is defined. Here, the function becomes undefinable when the denominator equals zero, which occurs when \( x - 3 = 0 \). Solving for \( x \) gives \( x = 3 \), meaning the function is not defined at \( x = 3 \). Thus, the domain in interval notation is \( (-\infty, 3) \cup (3, \infty) \). This means that every real number except 3 is part of the domain.

The range of \( f(x) = \frac{1}{x-3} \) includes all possible output values \( y \). For this function, because it has a horizontal asymptote at \( y = 0 \), \( y \) can take any real number except zero. Therefore, the range is described as \( (-\infty, 0) \cup (0, \infty) \). This means that every real number except zero can be a value of \( f(x) \). Understanding the domain and range is crucial for graphing and understanding the behavior of rational functions.

Asymptotes are lines that a graph of a function approaches but never actually touches. For rational functions like \( f(x) = \frac{1}{x-3} \), there are usually both vertical and horizontal asymptotes.

In this function, the vertical asymptote is located at \( x = 3 \). This is where the function becomes undefined and the values of \( f(x) \) grow toward infinity as \( x \) gets closer and closer to 3 from either side.

Horizontal asymptotes show the behavior of the graph as \( x \) approaches positive or negative infinity. For the function \( f(x) = \frac{1}{x-3} \), the horizontal asymptote is at \( y = 0 \). This indicates that as the value of \( x \) grows very large in either the positive or negative direction, the function approaches zero but never actually reaches it.

Recognizing these asymptotes helps to predict the behavior of the function near certain critical points and helps in accurately sketching the graph.

Interval notation is a mathematical way of representing sets of numbers, often used to describe the domain and range of functions.

In the context of our function \( f(x) = \frac{1}{x-3} \), we use interval notation to express the domain as \( (-\infty, 3) \cup (3, \infty) \). This notation signifies all real numbers except for 3. The symbol \((-) \) or \((+) \) next to infinity indicates that infinity is not a number but a concept representing unboundedness.

For ranges, interval notation helps describe all valid output values. Since the range of this function is \( (-\infty, 0) \cup (0, \infty) \), it implies all real numbers except 0. The use of unions (\( \cup \)) in interval notation helps indicate that there are two separate intervals involved.

Utilizing interval notation provides a clear and concise way to denote sets of numbers, especially when dealing with non-continuous functions like rational functions.