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Domain. When determining a domain, your are determining the permissible values of x. This raises the question: What is impermissible? Probably the first thing you learn that should never happen is: Dividing by zero. Your function has no divisions or denominators so this is not an issue here.
Another thing you learn should not happen (unless you want to work with complex numbers) is negative radicands of even-numbered roots. A radicand is the expression inside a radical. Even-numbered roots are 2nd roots, 4th roots, etc. And 2nd roots are better known as square roots. So this restriction is more popularly (but not as accurately) known as "no negative values inside a square root." Your function does have a square root so we need to make sure its radicand is not negative. IOW, we need to make sure the radicand is greater than or equal to zero. Translating this statement into a mathematical expression we get:
Finding a range can be more difficult. You have to have an understanding of how things work. Your function, in its essence, is the sum of 3 and a square root. Let's think about the square root part. Square roots are not negative. (If an expression needs to refer to the negative square root, it has to have a "-" in front of the square root.) And not only are they never negative, they can be 0 or any positive number So the lowest value your square root can be is zero. There is no highest value because there is no "highest" positive number. If your function was just this square root, then the range would be .
But your function is not just the square root. Your function is 3 plus the square root. If we take all the values that the square root can be and add 3 to them we should be able to see that the range would then be:
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