91Ó°ÊÓ

In mathematics, the domain of a function is a set containing all the possible inputs (or values) that can be used in the function. Think of it as the starting point for understanding how a function behaves. For logarithmic functions like the one given, the domain is slightly more particular. Because logarithmic functions are only defined for positive numbers, we must ensure that the argument (the content inside the log function) is greater than zero.
In our example, the function is \(f(x) = \log_5(x+4)\). The argument here is \(x+4\). To find the domain, we need to set \(x+4 > 0\). Solving this inequality gives us all the possible values for \(x\) that keep the log function valid and working.
Once we solve the inequality \((x+4 > 0)\), we find that \(x > -4\). Thus, the domain of the function, in this case, consists of all real numbers greater than \(-4\). We can express this set of numbers in a different, more concise format using interval notation.

Inequalities are expressions that describe the relative size of two values. Unlike equations, which state that two expressions are equal, inequalities show whether one side is less than, greater than, or not equal to the other side. They are crucial in solving problems related to domains, especially in logarithmic and other functions.
In the exercise we're discussing, we came up with an inequality \(x+4 > 0\). This inequality explains that, for the logarithm function to work, \(x+4\) must be more than zero.
To solve \(x+4 > 0\):

• Subtract 4 from both sides to isolate \(x\). This step gives us \(x > -4\).
• The solution shows that any value for \(x\) must be greater than \(-4\) to meet this condition.

Inequalities like this help us decide which values \(x\) can take, so the function remains applicable and doesn't run into mathematical issues, such as taking the logarithm of a non-positive number.

Interval notation is a simplified way to express a range of numbers. It helps mathematicians convey sets of numbers more succinctly, especially when dealing with continuous ranges, like domains of functions. This notation typically uses parentheses \(()\) and/or brackets \([]\).
For example:

• Parentheses, \(()\), indicate that an endpoint is not included in the set.
• Brackets, \([]\), mean the endpoint is included.

In the function \(f(x) = \log_5(x+4)\), after solving the inequality \(x > -4\), we discover that \(x\) can be any number greater than \(-4\), meaning \(-4\) itself is not included.
Thus, using interval notation, we write this domain as \((-4, +\infty)\). The parenthesis around \(-4\) shows that it is not part of the domain, aligning with our condition that \(x\) must be strictly greater than \(-4\). Interval notation simplifies communication of such ideas, making it clear to all mathematicians at a glance.