A quadratic function has a parabola as its graph; this graph changes direction at its vertex.

 exists if and only if, if , then - or, equivalently, if there does not exist  and  such that , but . This will happen on any interval on which the graph of  constantly increases or constantly decreases, but if the graph changes direction on an interval, there will be  such that  on this interval. The key is therefore to identify the interval that contains the vertex.

The -coordinate of the vertex of the parabola of the function

is .

The -coordinate of the vertex of the parabola of  can be found by setting :

.

Of the five intervals among the choices, 

so  cannot exist if  is restricted to this interval. This is the correct choice. 

 is an absolute value of a linear expression; as such, its graph is a "V" shape whose vertex occurs at the point at which .

 exists if and only if, if , then - or, equivalently, if there does not exist  and  such that , but . This will happen on any interval on which the graph of  constantly increases or constantly decreases, but if the graph changes direction on an interval, there will be  such that  on this interval. The key is therefore to identify the interval that contains the vertex.

Set

and solve for :

Of the four intervals in the choices, 

,

so  cannot exist if  is restricted to this interval. This is the correct choice.

Consider the function . Then 

.

  is an absolute value of a linear function. Since the value of this function cannot be less than 0, its graph changes direction at the value of  at which 

This value can be found by solving for :

, so the change of direction does not occur on the domain - it increases everywhere or decreases everywhere, so, if restricted to the given domain, we can treat this as if it were a simple linear function. Consequently, we can find the minimum and maximum at the endpoints. Evaluating  and :

This makes the range of   on the given domain.

In order to find the domain of a function, you need to state for what values of x the function can be true. The trick to finding the domain is to figure out what the value of x cannot be, and adjusting your answer appropriately.

In the function, , it is clear that x cannot be equal to 5. If x was 5, there would be a zero in the denominator, which would cause the function to be undefined. Remember, you can never have a zero in the denominator! That is the main thing to look out for when being asked for the domain of a function.

All other values for x are acceptable, so your answer is that x can be anything other than 5.

The proper way to write that x is any value other than 5 is,  

which states that the domain can be all numbers from negative infinity to positive infinity, excluding 5.

Consider the function . Then 

.

This is an absolute value function. Since the value of this function cannot be less than 0, its graph changes direction at the value of  at which . This value can be found by setting 

and solving for :

, so  assumes the value 0 on this domain. This must be the minimum value.

 A function of the form  is a linear function and is either constantly increasing or constantly decreasing. Therefore,  constantly increases on one side and decreases on the other. Therefore, the maximum must occur at either endpoint of the domain of . Evaluate both  and :

13, the greater of the two values, is the maximum value. The range of  on the given domain is .

The domain of a square root function is exactly the set of values of  for which the radicand takes on a nonnegative value. Therefore, it holds that to find the domain of , we can set up the inequality

and solve for  using the properties of inequality:

In interval notation, this is the set .

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 .