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 .  

Finding the Domain

The domain of a function is defined as the set of all valid input values of  overwhich the function is defined. The simple rule of thumb for rational functions is that all real numbers will work except for those in which denominator is zero since division by zero is not allowed.

Set the denominator to zero and solve for , 

The function is therefore defined everywhere except at . Therefore the domain expressed in interval notation is,

Note that the open parentheses indicate that  is not in the domain, but  may become arbitrarily close to  . 

Finding the Range 

The range of a function is defined as the set of all outputs spanning the domain. Finding the range can be achieved by finding the domain of the inverse function. First solve   for  to obtain the inverse function, 

Multiply both sides by , 

Distribute , 

Move all terms with  to one side of the equation, 

Factor and solve for 

The inverse function is therefore,

Find the domain of the inverse function, 

The range of  is the domain of , which is:

If you look at the plots for the function  (in blue) and  (in red and labeled as  in the figure) you can see the asymptotic behavior of as  approaches  and of  as  approaches .

To determine the domain of the function, we must ask ourselves where x can and cannot exist. On the numerator, nothing is preventing x from existing anywhere. But the denominator of the function cannot equal zero (which would produce an undefined value for the function), so to determine at which x values this occurs, we must set the denominator equal to zero and solve for x:

(The factors of 6 that add up to 5 are 3 and 2.)

Because these are the only values that x equal for the function to exist, we make our intervals as shown below:

We use round brackets to indicate that we never include the bounds of the intervals in the domain.

To determine the domain of the function, we must consider where x cannot exist. The only limitation on the function is the denominator, which cannot equal zero. 

To find the x-values where this occurs, we must set the denominator equal to zero and solve for x:

These are the only two limitations on the domain of the function, so the domain of the function is

Note that round brackets were used for all of the intervals, because none of the bounds of the intervals are included in the domain.