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: 

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in  values and plot points.

Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.

If we plug in -100000 for , we get 100000 for . 

If we plug in 100000 for , we get 100000 for .

Additionally, if we plug in any value for , we will see that we always get a real, defined value for .

**Extra Note: Due to the absolute value notation, the negative (-) next to the  is not important, in that it will always be made positive by the absolute value, making this function the same as  .  If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding -values negative. However, this knowledge is most important for range, rather than domain.

To find the range, plug each value of the domain into the equation:

As the x-values increase, the y-values do as well.  Therefore there is a  relationship

The range of a function is the group of corresponding  values for a given domain ( values).  Plug each  value into the function to find the range:

The range is .

Functions cannot have more than one  value for each value.  This means different numbers in the range cannot be assigned to the same value in the domain.  Therefore, cannot be the domain of the function.

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in  values and plot points.

Upon doing so, we learn that the -values (range) never surpass . This is because of the negative that is placed outside of the absolute value function. Meaning, for every  value we plug in, we will always get a negative value for , except when .

With this knowledge, we can now confidently state the range as 

**Extra note: the negative sign outside of the absolute value is simply a transformation of , reflecting the function about the -axis. 

 Finding the Domain

The domain of a function is the set of all  over which the function  is defined. 

 When adding functions the domain is the intersection between the domains of the two functions. In our case we will consider  to be the sum of two funcitons  and . The domain of  is simply all real numbers , so the domain of  will be whatever the domain of  . 

The domain of  can be found by remembering that everything under the radicand must be either a postive real number or zero. Apply this condition, 

So the domain is, 

Here we can see graphically what happens when we add the fucntions. The red line is , and the green curve is  . The blue line is the sum of these two fucntions , and has the same domain as the function in green which starts at , and continues for all real numbers greater than .