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Understanding the domain of a function is crucial because it tells us the set of all possible input values (x-values) for which the function is defined. For the function \( g(x) = \frac{1}{2} \sqrt{x} \), we need to keep in mind that the square root function \( \sqrt{x} \) is only defined for non-negative numbers.

This means the domain of \( g(x) \) is all values of \( x \) such that \( x \geq 0 \). In other words, \( x \) can be zero or any positive number. If we try to plug in a negative number for \( x \), the square root is not a real number, and the function doesn't produce a valid output.

To summarize:

• The domain of \( g(x) = \frac{1}{2} \sqrt{x} \) is all non-negative numbers: \( [0, \infty) \)

Creating a table of values is a simple way to understand how the function behaves. By choosing specific \( x \) values, calculating the corresponding \( g(x) \) values, and listing them, we can see the relationship between \( x \) and \( g(x) \).

Let's select some convenient x-values and compute \( g(x) \):

• When \( x = 0 \), \( g(0) = \frac{1}{2} \sqrt{0} = 0 \)
• When \( x = 1 \), \( g(1) = \frac{1}{2} \sqrt{1} = \frac{1}{2} \)
• When \( x = 4 \), \( g(4) = \frac{1}{2} \sqrt{4} = 1 \)

By listing these points, we create a reference that we can use to plot the function on a graph. Here’s what our table of values looks like:

• \( (0, 0) \)
• \( (1, \frac{1}{2}) \)
• \( (4, 1) \)

Once we have our table of values, we can start plotting points on a coordinate plane. This step helps us visualize how the function behaves.

Take the points from our table of values:

• \( (0, 0) \)
• \( (1, \frac{1}{2}) \)
• \( (4, 1) \)

Plot these points on the graph, with \( x \)-coordinates on the horizontal axis and \( y \)-coordinates on the vertical axis. After plotting the points, draw a smooth curve that connects them.

This curve represents the graph of the function \( g(x) = \frac{1}{2} \sqrt{x} \). The graph should start at the origin \( (0,0) \) and rise slowly to the right, indicating that as \( x \) increases, \( g(x) \) also increases, but at a slower rate.

By following these steps, we can clearly see how the function behaves and understand its graphical representation.