The domain of a function is the set of possible values for the variable. The range would be the possible values for the solution, .

The value inside of a square root must be greater than or equal to zero in order to have a real solution.

Now we can solve for in the inequality.

Square roots have positive and negative roots, so we need to set up two results. Remember to switch the direction fo the inequality for the negative solution.

These solutions can be combined to give our final answer.

Any values of that are not included will result in an imaginary (impossible) answer.

The domain of a function includes all of the values of x for which f(x) is real and defined. In other words, if we put a value of x into the function, and we get a result that isn't real or is undefined, then that value won't be in the domain.

If we let x = 0, then we will be forced to evaluate , which is equal to 1/0. The value of 1/0 is not defined, because we can never have zero in a denominator. Thus , because f(0) isn't defined, 0 cannot be in the domain of f(x).

The answer is 0.

The domain of a function consists of all of the possible values for x. In this case, we want to make sure that we are not dividing by  ( in the denominator), since that would make our function undefined. Having  would make the denominator . Thus, that is not in our domain. There is nothing else that would make this function undefined, and thus the domain is all real numbers except

The range of a function is all of the possible values that the equation can take. For this equation our value cannot be negative, as a negative number squared still gives us a positive value.

Since , we know that the lowest possible value that can reach is . Therefore the range is .

Domain represents all possible values for . In this situation, we want to make sure that the numbers under the square root are greater than or equal to 0. This will be the case when . Thus, the domain includes all real numbers such that . 

The range of a function is defined as the possible values for , or the possible outcomes. In this function, it is not possible to get any sort of negative number as an outcome. It is possible to get zero when . Thus, the range is all real numbers greater than or equal to 0. 

The domain includes the values that go into a function (the x-values) and the range are the values that come out (the  or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum  is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is .

Using  as the input () value for this equation generates an output () value that contradicts the stated condition of .

Therefore  is not a valid value for  and not in the equation's domain: