Problem 12 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, 0)\).

Step 1: Understanding the Logarithmic Function

The function given is \( y = \log\left(-\frac{1}{2}x\right) \). For the logarithm function \( \log(a) \) to be defined, the argument \( a \) must be greater than zero. Therefore, we need to ensure the argument \(-\frac{1}{2}x > 0 \).

Step 2: Inequality Setup

Set up the inequality that needs to be satisfied for the logarithm to be defined. Start with the expression within the logarithm: \(-\frac{1}{2}x > 0\).

Step 3: Solving the Inequality

Solve the inequality \(-\frac{1}{2}x > 0\). First, multiply both sides by \(-1\), which reverses the inequality sign: \(\frac{1}{2}x < 0\). Then, multiply both sides by \(2\) to isolate \(x\): \(x < 0\).

Step 4: Conclusion about the Domain

The solution \(x < 0\) indicates that for the logarithmic function to be defined, \(x\) must be less than zero. Therefore, the domain of the function \(y = \log\left(-\frac{1}{2}x\right)\) is all negative real numbers: \((-\infty, 0)\).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical operations that are the inverse of exponentiation. They are of the form \( y = \log_b(x) \), where \( b \) is the base of the logarithm (often 10 for common logarithms or \( e \) for natural logarithms) and \( x \) represents the argument. The logarithm \( \log_b(x) \) is only defined for arguments \( x > 0 \). This constraint emerges because only positive numbers are reasonable results of exponentiation with real number bases. Logarithmic functions are continuous within their domain and possess a unique property—values grow very slowly compared to exponential functions. These aspects make them useful in fields like science and engineering where large ranges of values must be analyzed.

Inequalities

Inequalities are expressions that show the relationship between two quantities, indicating whether one is greater, less than, or not equal to another. When dealing with logarithmic expressions, setting up inequalities is crucial to understanding their domain. For the function \( y = \log(-\frac{1}{2}x) \), we must solve the inequality \(-\frac{1}{2}x > 0\) to determine when the logarithm is defined. By isolating \( x \) through operations while remembering that multiplying or dividing by a negative number flips the inequality sign, we can solve it: \( x < 0 \). This result shows that the domain is all values less than zero, ensuring that the argument stays positive as required in a logarithmic function.

Graphical Support

Graphical support can greatly aid in comprehending abstract mathematical concepts. For this function, plotting its graph helps visually confirm the domain and behavior of the function \( y = \log(-\frac{1}{2}x) \). The graph of the logarithm typically has a vertical asymptote along the y-axis, where the function is undefined. Since our function involves a negative coefficient in the argument, we would expect the graph to show that the function shifts and reflects along the y-axis, confirming that it is only defined for negative \( x \)-values. Such visual cues allow for a deeper understanding of how transformations affect the graph's standard shape and provide reinforcement of the analytical findings regarding the function's domain.

Function Analysis

Function analysis involves examining the properties of functions to understand their behavior. This can include investigating domain, range, asymptotes, and other important features. For the logarithmic function \( y = \log(-\frac{1}{2}x) \), a detailed analysis starts by identifying its domain, \( x < 0 \). From there, one can explore other aspects like intercepts, albeit this function does not have an x-intercept as it's only defined for \( x < 0 \). The function's range is all real numbers since a logarithmic function can output any real value when the argument is defined. Asymptotic behavior is another crucial part of the analysis, revealing how the function nears certain lines without actually reaching them—here, the function nears the y-axis but never crosses it. Comprehensive function analysis allows students to grasp how every part of the function interacts and addresses broader mathematical inquiries.