The domain of a rational function is the set of all real numbers except for the value(s) of that make the denominator zero. The value(s) can be found as follows:

However, there is no real value of whose square is , so this statement has no solution. Therefore, there is no real value of which makes the denominator zero. The domain of is consequently the set of all real numbers.

is a polynomial function. The domain of any polynomial function is the set of all real numbers, making that the correct choice.

The domain of a square root function is the set of all values of for which the radicand is nonnegative. Three of these functions are undefined for at least one value of , since this value yields a negative radicand. 

 is outside the domain of , , and by virtue of causing their radicands to be negative:

:

We examine . We show that the radical can never be negative by setting it as such, and trying to solve for , as follows:

Since must be nonnegative, it is always greater than . Therefore, the inequality has no solution. Consequently, the radicand of is always nonnegative, and has the set of all real numbers as its domain.

This function resembles a parabola since the highest order is within the term .

There are no denominators where the  variable is undefined.  

The domain refers to the existing x-values which lie on the graph.

This parabola will only shift upward eight units and will not affect the domain.

The answer is:  

The easiest way to figure this out is by knowing that the domain of all quadratics without any restrictions is always all real numbers, though this problem can be solved graphically, too.

Either method is correct and your correct answer is:

It is important, too, to note that soft brackets must be used when working with infinity. 

There are two important components of this problem. First we must set the denominator equal to zero and solve for . This gives us values that  can't be because the denominator can never equal zero.

Doing this, we get , so,  can be any number except for a positive two.

The other key to this problem is knowing the difference between brackets and parentheses. The square brackets mean that a number is included, whereas the parentheses mean that it is not. Because  cannot equal 2, we need to use the parentheses.

This makes:

From the figure one can see the Range varies from  to .

Let  be the mass of the weight and the elongation of the spring. Then for some constant of variation , 

We can find  by setting  from the first situation:

so 

In the second situation, we set  and solve for :

which rounds to 11.5 centimeters.

If  varies directly with the square root of , then for some constant of variation , 

If , then ; therefore, the equation becomes 

, 

or

.

Divide by 5 to get , making the equation 

.

If , then .

Let  be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation , 

.

We can find  by setting :

Therefore .

Set  and solve for :

 kilograms