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

, so the vertex is not on the domain.  The maximum and the minimum of  must occur at the endpoints, so evaluate  and .

The minimum and maximum values of  are  and 40, respectively, so the correct range is .

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

, so the vertex is on the domain.  The maximum and the minimum of  must occur at the vertex and one endpoint, so evaluate , , and .

The minimum and maximum values of  are  and 17, respectively, so the correct range is .

First, we must determine whether  exists.

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 determine whether the interval to which the domain is restricted 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 :

.

The vertex of the graph of  without its domain restriction is at the point with -coordinate . Since , the vertex is not in the interior of the domain; as a consequence,  exists on .

To determine the inverse of , first, rewrite  in vertex form

, the same as  in the standard form.

The graph of , if unrestricted, would have -coordinate , and -coordinate 

Therefore, .

The vertex form of  is therefore

Replace  with :

Switch  and :

Solve for . First, add 46 to both sides:

Multiply both sides by 2:

Take the square root of both sides:

Subtract 8 from both sides

Replace  with :

Either  or 

The domain of  is the set of nonnegative numbers; this is consequently the range of .  can only have negative values, so the only possible choice for  is .

A quadratic function has a parabola as its graph; this graph changes direction at its 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  and :

.

Since , the vertex falls within the domain of . 

Since, in , the quadratic coefficient is positive, the parabola curves upward. On the set of all nonnegative numbers, the function has no maximum. The minimum occurs at the vertex, which is in the domain; to calculate it, evaluate :

The range of the function given the domain restriction is .

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 .