91Ó°ÊÓ

An exponential function is a mathematical function in the form of \( f(x) = a \, b^{x} \), where \( a \) is a constant, and \( b \) is a positive real number known as the base. The function has interesting properties due to the way it grows or decays over different input values. Exponential growth occurs when \( b > 1 \), meaning that as \( x \) increases, the value of the function increases rapidly. In contrast, exponential decay happens when \( 0 < b < 1 \), causing the function value to decrease as \( x \) rises.The function \( f(x) = \frac{3}{1 - e^{2x}} \) includes an exponential component \( e^{2x} \), where \( e \) is a special base around 2.718 called Euler's number. The exponent part \( 2x \) influences how fast the function grows or shrinks as \( x \) changes. By understanding these characteristics, students can predict behaviors and transformations of similar functions effectively.

Function notation is a way of representing functions in mathematics that clarifies the relationship between the input and the output. It is denoted as \( f(x) \), where \( f \) is the name of the function and \( x \) is the variable or input value.This notation is essential as it provides a clear format to communicate which specific function is being discussed and allows us to substitute any number for \( x \) to find corresponding outputs. For example, in the function \( f(x) = \frac{3}{1 - e^{2x}} \), replacing \( x \) with any real number except 0 will give us the value of the function at that point.Using function notation helps in:

• Identifying and evaluating functions quickly.
• Understanding the inputs (dominos) and corresponding outputs (range) clearly.
• Simplifying the process of composing or decomposing functions by using substitutions.

By mastering function notation, the interaction with both simple and complex functions becomes more intuitive and easier to manage.

Interval notation is a concise way to describe the domain or range of a function. It includes intervals made up of numbers, demonstrating where a function is defined or takes on values.For instance, with the function \( f(x) = \frac{3}{1 - e^{2x}} \), the domain in interval notation is \((-\infty, 0) \cup (0, \infty)\). This notation signifies all real numbers except for 0, showing that the function is undefined when \( x = 0 \).Key features of interval notation include:

• The use of parentheses \(()\) shows values not included in the interval, indicating that the endpoint is not part of the set.
• Brackets \([]\) mean values are included in the interval, marking endpoints as part of the set.
• Union symbol \(\cup\) combines separate intervals into one, especially when parts are disconnected, as with domain excluding specific points.

Learning interval notation allows students to express mathematical ideas precisely and is useful for conveying important concepts in calculus and beyond.