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When working with functions, the domain refers to all the possible input values that allow the function to work properly. For logarithmic functions like the one we have, which is given by \( y = \log \left( \frac{1}{2} x \right) \), the domain consists of values for which the inside of the logarithm is positive. This is because you can't take the logarithm of zero or a negative number in the realm of real numbers.
To find the domain of the function, we focus on ensuring the expression inside the logarithm, \( \frac{1}{2} x \), stays greater than zero. We set up an inequality \( \frac{1}{2} x > 0 \) and solve this to find \( x > 0 \). Thus, the domain of the function is all positive real numbers: \( (0, \infty) \).

Inequalities are mathematical expressions involving the symbols \( >, <, \geq, \leq \), reflecting the relations between two values. In the context of logarithmic functions, identifying inequalities helps determine the domain by making sure the expression inside the log is positive.
For the function \( y = \log \left( \frac{1}{2} x \right) \), we used the inequality \( \frac{1}{2} x > 0 \). By solving this inequality, we learn that \( x \) must be greater than zero to ensure the argument inside the log is positive.
Solving inequalities often involves similar steps to solving equations, such as multiplying both sides of the inequality to eliminate fractions. However, be cautious—especially with inequalities involving multiplying or dividing by negative numbers—as these actions would flip the inequality sign. Fortunately, in our example,we only multiplied by a positive number (2), maintaining \( x > 0 \).

Graphical representation of functions helps visualize potential solutions. While not always necessary to find the domain analytically, checking the graph of the function \( y = \log \left( \frac{1}{2} x \right) \) can provide a visual confirmation of the domain found earlier.
If you were to graph this function, you would note that the curve only appears on the right side of the y-axis, matching the analytical domain \( x > 0 \). As you observe the graph, it becomes evident that the function does not exist at \( x = 0 \) or negative values, reinforcing the finding that \( x \) must be positive.
Visual tools like graphing are excellent for ensuring calculations align with function behavior, as the graph will clearly show for which x-values the function produces real, defined outputs. This visual insight gives confidence and a deeper understanding beyond analytical calculation alone.