91Ó°ÊÓ

The natural logarithm, denoted as \( \text{ln}(y) \), is an important mathematical function. It is the inverse of the exponential function \( e^y \). This means if \( \text{ln}(y) = a \), then \( y = e^a \). The natural logarithm is only defined for positive numbers, which means that its argument must always be greater than zero.

For instance, consider the function in the problem: \( g(x) = 8 + 5 \text{ln}(2x + 3) \). Here, \( 2x + 3 \) must be positive because the natural logarithm cannot process non-positive values. This leads us to our next concept, setting up an inequality to find the valid domain.

An inequality is a mathematical statement that indicates the relationship between two expressions. It shows whether one expression is smaller, greater, or simply not equal to another. To solve our problem, we need to set up an inequality based on the fact that the argument of the natural logarithm must be greater than zero.

For \( g(x) = 8 + 5 \text{ln}(2x + 3) \), we establish the inequality \( 2x + 3 > 0 \). Solving this step by step:

• Subtract 3 from both sides: \( 2x > -3 \)
  
  
• Divide both sides by 2: \( x > -\frac{3}{2} \)

By solving this inequality, we determine that \( x \) must be greater than \( -\frac{3}{2} \) for the function \( g(x) \) to be defined. This directly helps in finding the domain of the function.

Interval notation is a method of writing sets of numbers, often used to describe the domain or range of a function. It uses parentheses \(()\) and brackets \([]\) to show where intervals start and end. Parentheses mean the endpoint is not included, while brackets mean the endpoint is included.

In our example, we've found that \( x \) must be greater than \( -\frac{3}{2} \). To express this in interval notation, we use an open parenthesis to show that \( -\frac{3}{2} \) is not included:

The domain of \( g(x) = 8 + 5 \text{ln}(2x + 3) \) is written as \( (-\frac{3}{2}, \text{infinity}) \). This notation signifies that \( x \) can take any value greater than \( -\frac{3}{2} \) and extends infinitely in the positive direction. Understanding interval notation is crucial for writing the domain and range of functions clearly and accurately.