The domain is the set of possible value for the variable. We can find the impossible values of by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.

Now we can solve for .

There is no real value of that will fit this equation; any real value squared will be a positive number.

The radicand is always positive, and is defined for all real values of . This makes the domain of  the set of all real numbers.

To find the domain, find all areas of the number line where the fraction is defined.

because the denominator of a fraction must be nonzero.

Factor by finding two numbers that sum to -2 and multiply to 1.  These numbers are -1 and -1.

The domain is the set of x-values that make the function defined.

This function is defined everywhere except at , since division by zero is undefined.

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:

This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "[" for the beginning.

Minimum: 5 inclusive, maximum: infinity

Range: 

The domain represents the acceptable  values for this function. Based on the members of the function, the only limit that you have is the non-allowance of a negative number (because of the square root). The square and the linear terms are fine with any numbers. You cannot have any negative values, otherwise the square root will not be a real number.

Minimum: 0 inclusive, maximum: infinity

Domain: 

The domain of a function refers to the viable  value inputs. Common domain restrictions involve radicals (which cannot be negative) and fractions (which cannot have a zero denominator).

This function does not have any such restrictions; any value of  will result in a real number. The domain is thus unlimited, ranging from negative infinity to infinity.

Domain: 

This function represents a parabola that has been shifted 15 units to the left and 2 units up from its standard position.

The vertex of a standard parabola is at (0,0). By shifting the graph as described, the new vertex is at (-15,2). The  value of the vertex represents the minimum of the range; since the parabola opens upward, the maximum will be infinity. Note that the range is inclusive of 2, so you must use a bracket "[".

Minimum: 2 (inclusive), maximum: infinity

Range: 

The domain of a function refers to the viable  value inputs. Common domain restrictions involve radicals (which cannot be negative) and fractions (which cannot have a zero denominator). Both of these restrictions can be found in the given function.

Let's start with the radical, which must be greater than or equal to zero:

Next, we will look at the fraction denominator, which cannot equal zero:

Our final answer will be the union of the two sets.

Minimum: 2 (inclusive), maximum: infinity

Exclusion: 22

Domain: 

There are two limitations in the function: the radical and the denominator term. A radical cannot have a negative term, and a denominator cannot be equal to zero. Based on the first restriction (the radical), our  term must be greater than or equal to zero. Based on the second restriction (the denominator), our  term cannot be equal to 4. Our final answer will be the union of these two sets.

Minimum: 0 (inclusive), maximum: infinity

Exclusion: 4

Domain: