4. \( y=\sqrt{x} \) a \( x \geq 0 \) (b) \( \{x \in \mathbf{R} \mid x \neq 0\} \) (c) \( \{x \in \mathbf{R} \mid x>0\} \) (d) \( \{x \in \mathbf{R} \mid x \leq 0\} \)

Did you know that the equation \( y = \sqrt{x} \) has quite a storied history dating back to ancient civilizations? Mathematicians in ancient India and Greece were already exploring the properties of square roots, laying the groundwork for modern algebra. The use of \( y = \sqrt{x} \) symbolizes the pursuit of understanding relationships between variables, revealing how changes in \( x \) directly influence the output \( y \), especially in geometry where the graph of this function is a classic parabolic shape! As you work with this function, be cautious of some common pitfalls, especially when considering its domain. For instance, choosing \( x < 0 \) can lead to errors, as the square root of negative numbers isn't defined in real numbers. Always ensure you analyze the given sets carefully—while the natural domain of \( y = \sqrt{x} \) is \( x \geq 0 \), attempting to take the square root of negative values will lead to complex numbers, completely changing the context of your problem!