You know that the square root function has real roots only for positive values. Therefore, you know that . From this, you can solve for your domain:

For this function, the only thing to which you need to pay heed is the denominator of the fraction, which cannot be equal to zero. Thus, you can just look at the polynomial found in the denominator and solve for the values of  equal to

This is solved by factoring:

This indicates that 

Otherwise, all values are fine!

The domain is the set of all  where the function is defined. Since we can't divide by 0, we need to find the  values that would make the denominator zero. Set the denominator equal to zero and solve for  to find:

Thus  can take all values except for positive and negative 2, so the domain is:

The domain of the function  is , because the result of taking the square root of a negative number is imaginary.

Therefore, for the function given, ,  when .

So the domain must be .

To find the domain we want to find any areas for which an x value will not work in our function. The potential problem area would be that under the radical.

Therefore, for the function given, ,  for all real numbers .

So the domain must be .

In other words, there is no real value of  whose absolute value is less than  let alone 

Looking at our numerator shows us that  can never be negative, as it would result in a nonreal solution for the numerator. The denominator solves to a real value for all positive integers until , where it causes us to divide by zero, and all numbers past  result in a negative radical (and thus a nonreal solution) for the denominator.

Testing  itself produces , which is a fine answer. Therefore, our domain is inclusive of  and exclusive of , and should be written as .

The two potential concerns for domain in a standard function are negative numbers inside even-powered radicals, and dividing by zero. In this case, the radical we have is odd-powered, so having a negative result underneath the radical is fine. All we need to do, then, is avoid dividing by zero, which means avoiding a result which sums to zero under the radical. In this case, only  will create this situation, so we must avoid it.

Thus, our domain is 

To find the domain, you must find the values for which you can plug in for . Based on the given function, you know that you can not have a demoninator equal to . So, by setting the denominator equal to , you can find out the values that can not be in the domain, and then simply remove them from your set of numbers from negative infinity to positive infinity.

Thus,

Therefore, for our function,  can not equal  or it will be undefined. Thus, we get the set of numbers excluding positive .

 has an inverse on a given domain if and only if there are no two distinct values on the domain  such that .

The key to this question is to find the zeroes of the polynomial, which can be done as follows:

'

The zeroes are . 

 has one boundary that is a zero and one interior point that is a zero. Therefore, there is a vertex in the interior of the interval, so it will have at least one pair  such that . Since a cubic polynomial has two "arms", one going up and one going down,  will increase as  increases in the other four intervals.  is the correct choice.

 has an inverse on a given domain if and only if there are no two distinct values on the domain  such that .

 has a sinusoidal wave as its graph, with period  and phase shift  units to the left. Its positive "peaks" and  "valleys" begin at  and occur every  units.

Since  includes one of these "peaks" or "valleys", it contains at least two distinct values  such that . It is the correct choice.