The range of a function is the set of y-values that a function can take. First let's find the domain. The domain is the set of x-values that the function can take. Here the domain is all real numbers because no x-value will make this function undefined. (Dividing by 0 is an example of an operation that would make the function undefined.)

So if any value of x can be plugged into y = x² + 2, can y take any value also? Not quite! The smallest value that y can ever be is 2. No matter what value of x is plugged in, y = x² + 2 will never produce a number less than 2. Therefore the range is y ≥ 2.

Values of x that make the denominator equal zero are not included in the domain. The denominator can be simplified to (x – 3)², so the value that makes it zero is 3.

The domain of a relation is the same as the range of the inverse of the relation.  In other words, the x-values of the relation are the y-values of the inverse.

The range of a function is the set of y-values that a function can take. First let's find the domain. The domain is the set of x-values that the function can take. Here the domain is all real numbers because no x-value will make this function undefined. (Dividing by 0 is an example of an operation that would make the function undefined.)

So if any value of x can be plugged into y = x² + 2, can y take any value also? Not quite! The smallest value that y can ever be is 2. No matter what value of x is plugged in, y = x² + 2 will never produce a number less than 2. Therefore the range is y ≥ 2. 

We need to be careful here not to confuse the domain and range of a function. The problem specifically concerns the range of the function, which is the set of possible numbers of . It can be helpful to think of the range as all the possible y-values we could have on the points on the graph of .

Notice that  has  in its equation. Whenever we have an absolute value of some quantity, the result will always be equal to or greater than zero. In other words, |4-x| 0. We are asked to find the smallest value in the range of , so let's consider the smallest value of , which would have to be zero. Let's see what would happen to  if .

This means that when , . Let's see what happens when  gets larger. For example, let's let .

As we can see, as  gets larger, so does . We want  to be as small as possible, so we are going to want  to be equal to zero. And, as we already determiend,  equals  when .

The answer is .

 is the same as . 

To find the inverse simply exchange  and  and solve for . 

So we get  which leads to .

The domain of the inverse of a relation is the same as the range of the original relation. In other words, the y-values of the relation are the x-values of the inverse.

For the original relation, the range is: .

Thus, the domain for the inverse relation will also be .

The domain means what real number can you plug in that would still make the function work. For this case, we have to worry about the denominator so that it does not equal 0, so we solve the following. 7x – 1 = 0, 7x = 1, x = 1/7, so when x ≠ 1/7 the function will work.