Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$f(x)=\log \left(-\frac{1}{2} x\right)$$

Step-by-Step Solution

Answer

The domain is all x such that x < 0, or (-∞, 0) in interval notation.

Step 1: Understand the Domain of Logarithmic Functions

For a logarithmic function \ f(x) = \log_a(bx) \, where \( a \) is the base and \( bx \) is the argument, the domain is defined where the argument is positive. This means that \( bx > 0 \). We will apply this concept to determine the domain of \ f(x) = \log \left(-\frac{1}{2} x\right) \.

Step 2: Set Up the Inequality for the Argument

Set the argument of the logarithmic function greater than zero: \(-\frac{1}{2} x > 0\). This inequality will help us find the possible values of \( x \) that are within the domain of the function.

Step 3: Solve the Inequality

To solve the inequality \(-\frac{1}{2} x > 0\): \ 1. Multiply both sides by \(-2\) to isolate \( x \), and remember to reverse the inequality sign: \( x < 0\). Thus, the solution to this inequality is any real number less than zero.

Step 4: Write the Domain

The domain of the function \ f(x) = \log \left(-\frac{1}{2} x\right) \ is all \( x \) such that \( x < 0 \). Therefore, the domain in interval notation is \ (-\infty, 0) \.

Key Concepts

Domain of a Function

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.

Inequality Solving

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

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.