What are the domain and range of the real-valued function ?
The domain is , and the range is .
The domain is , and the range is .
The domain is , and the range is all real numbers.
The domain is , and the range is .

To find the domain of the function , we need to ensure the expression under the square root is non-negative. Thus, we set , leading to . So, the domain is .

Now for the range, we note that the smallest value that can take when is . Therefore, . As increases, also increases, meaning can grow indefinitely. Hence, the range is .

So the correct answer is: The domain is , and the range is .

Do you know that square roots have a fascinating history? They were first used by the ancient Babylonians around 2000 BCE! They developed methods for approximating square roots, long before modern algebra emerged. Such a clever bunch to tackle such complex concepts so early!

If you’re looking to dive deeper into the world of functions and their properties, consider exploring the works of mathematicians like Descartes or even learning about how functions are used in programming for coding algorithms! Understanding these concepts will sharpen your problem-solving skills and you’ll impress your friends with your mathematical prowess.