The largest value that the function can have is when the square root part of the function is zero (it cannot be smaller than zero).

The largest value is 3, so the range must be 

The domain of any quadratic function is always all real numbers.

The range of this function is anything greater than or equal to 5.

These written in the correct notation is:

Soft brackets are needed for infinity and a hard square bracket for 5 because it is included in the solution.

To figure out the values that x (your domain) can't be you need to set the denominator equal to 0 since you cannot have 0 in the denominator.

When you solve this you get 

This means that you have have all values of x other than -5.

This notation of this is:

This function has the domain of x can't be 0 as well as the range of y can't be 0.

This means that the domain and range of these functions include everything but 0, which is denoted as:

 and  

Upon solving this inequality you get

This means that "y" is all values including 18 and greater.

Written as interval notation you get:

The trick here is to factor the top. Upon doing so you get:

From here you can cancel your "x+3"s to get 

  which has a domain of  

The domain of this function - the possible x values - is not limited at all by the numerator of the fraction, but by the denominator. When the denominator of a fraction is zero, this value is undefined.

To determine where x cannot exist, we must set the denominator equal to zero and solve for x:

The only restriction on x is at this value, so our domain is

 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 .  

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 range of the function, we must ask ourselves where the natural logarithm is valid (what y values can it produce). The natural logarithm is never negative, so the lowest value it reaches is zero. The function has a positive one added to it, so the lowest y-value the function reaches is 1. The function has no upper bounds, so it goes to infinity.