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.

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.

To determine the domain of the function, we must find the x-values that would give us an undefined result when we plug them into the function. On the numerator, we know that the natural log function can never equal zero or be negative. In the denominator, we can never have a zero. With these conditions in mind, we must now find the x values that makes these undefined situations occur.

For the numerator:

 and 

For the denominator:

Now that we know where x cannot be, we can now write the domain, making sure to use round brackets for the endpoints of the intervals:

To find the domain of the function, we must ask ourselves where x cannot be if we want the function to be defined. 

The only limitation on the function is that we can't have zero in the denominator (this is undefined). To find the x value(s) where this occurs, we set the denominator equal to zero and solve for x:

This is the only x value that we have an undefined result for the function, so our domain is

.

Note that we use no square brackets, because we are not including the endpoints of the intervals in the domain.

Range asked for the values of y expressed by the function. The absolute value function is always positive with respect to y. The point 0 is included in the range requiring a square bracket. Infinity is not a numerical value so it is bound by a parentheses. 

Domain is finding the acceptable  values that will make the function generate real values.  is a linear function therefore any  values will always generate real values. Answer is all real numbers.

Domain is finding the acceptable  values that will make the function generate real values.  is a quadratic function therefore any  values will always generate real values. Answer is all real numbers.