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Understanding the domain of a function is crucial when studying different kinds of functions, such as logarithmic functions. The domain refers to all possible input values

• Here, we look at values of \( x \) that can be put into the function to give feasible or real outputs.

For a logarithmic function like \( f(x) = \log(-\frac{1}{2} x) \), it’s important to recognize that you can only take the logarithm of a positive number. This means that the part inside the log (called the "argument") must always be positive.
To find the domain of \( f(x) = \log(a \cdot x) \), set up the inequality \( a \cdot x > 0 \). Solving this inequality gives us the range of values for \( x \) that maintain a positive argument. So, in this specific function, \(-\frac{1}{2} x\) must be positive, leading us to discover the domain where \( x < 0 \).
This process ensures the logarithmic function is defined and shows how we circumvent any possibility of taking a log of a negative or zero.

Solving inequalities is a vital math skill, particularly when dealing with domains of functions. Inequalities describe a range of possible values, unlike equations that pinpoint exact values. For instance, consider the inequality \(-\frac{1}{2} x > 0\).

• To solve, a common yet crucial step is multiplying or dividing both sides to isolate the variable \( x \).

One must remember the rule: whenever you multiply or divide by a negative number, the inequality sign flips. Here’s how you work through it:
1. Multiply each side of the inequality \(-\frac{1}{2} x > 0\) by \(-2\) (keeping in mind the sign reversal):
2. This transforms the inequality into \( x < 0 \).
This leads us to the solution, representing all real numbers less than zero. Such understanding of inequalities is essential when defining valid inputs for functions.

Interval notation is a clear and concise way of writing the set of solutions for inequalities, representing the domain of functions. It uses a pair of numbers in parentheses or brackets to show where an interval begins and ends. For our function with \( x < 0 \), the solution set of the inequality \(-\frac{1}{2} x > 0\) is recorded like this:

• Because \( x \) must be less than zero with no specified lower boundary, we start with \(-\infty\) and use a parenthesis.
• Since zero is not included in the value of \( x \) that makes the inequality true, close with \( 0) \) also with a parenthesis.

Thus, the interval notation for the domain is \((-\infty, 0)\). This format quickly conveys that the domain consists of all real numbers less than zero, making it a standard tool in conveying domain or solution sets in mathematics.