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A square root function is a type of function in which the variable is inside a square root. The general form of a square root function is \(f(x) = \sqrt{g(x)}\). The key feature is that the expression inside the square root, \[g(x)\], needs to be non-negative because the square root of a negative number isn't a real number.

Here, the given function is \(f(x) = \sqrt{\frac{1}{2}x + 3}\). In this case, \[g(x) = \frac{1}{2}x + 3\] needs to be non-negative for the square root to be defined. This leads us to setting up an inequality to find the domain of the function.

To ensure the square root function \(f(x) = \sqrt{\frac{1}{2}x + 3}\) is defined, the expression inside the root, \[g(x) = \frac{1}{2}x + 3\], must be non-negative. We write this as: \[\frac{1}{2}x + 3 \geq 0\]

Solving this inequality involves a few algebraic steps:

• Subtract 3 from both sides: \[\frac{1}{2}x \geq -3\]
• Multiply both sides by 2 to isolate x: \[x \geq -6\]

These steps result in the inequality \[x \geq -6\]. Thus, any value for x that is greater than or equal to -6 will make the function defined. Additionally, this inequality helps us to determine the set of all possible x-values that ensure the function produces real numbers, helping us find the domain of the function.

The domain of a function is the set of all possible input values (x-values) that make the function defined and produce real numbers. For the given function, \(f(x) = \sqrt{\frac{1}{2}x + 3}\), we determined through solving the inequality, \[\frac{1}{2}x + 3 \geq 0\], that the domain is \[x \geq -6\].

This means the function is defined for any x-value that is -6 or greater. In interval notation, the domain can be written as \[[-6, \infty)\]. So, \(-6\) is the smallest x-value for which the function is defined, and the function can take any larger x-value.

Understanding the domain of a function is crucial in mathematical analysis, as it tells us the scope of input values for which the function will produce meaningful outputs.