Domain is finding the acceptable  values that will make the function generate real values. Because this function is a fractional expression, we need to check the denominator. Remember, the denominator must not be zero. This would make the function undefined. 

 We only need to look at . Since we know it can't equal zero, we set that expression to zero to determine the -value that makes this function undefined.

 Add  on both sides.

 Take the square root of both sides. Remember it can also be negative.

Domain is finding the acceptable  values that will make the function generate real values. Because we have a radical, we have to remember the smallest possible value inside the radical is zero. However, because there's a fraction inside the radical, we should focus on that first. Remember, the denominator must not be zero. This would make the function undefined. 

 Subtract  on both sides.

 Let's check values greater than this for the radical function. Let's pick .

 This is good as this value is greater than zero. Let's pick a value less than  such as .

 This is not good as this value is less than zero and that's not acceptable in radical conditions. Therefore our answer will be . Remember  is not included. This makes the denominator zero and the whole function becomes undefined.

Because the domain is giving us a wide range of  values, we can easily eliminate the fractional function  as it only isolates a single  value. We can eliminate  as it means I am restricted to  as my domain but I am looking for domain values greater than . This leaves us with the radical functions.

We have to remember the smallest possible value inside the radical is zero. Anything less means we will be dealing with imaginary numbers.

 ,  This means the domain is  which doesn't match our domain so this is wrong. 

 , . This means the domain is  which doesn't match our domain since we want to EXCLUDE  so this is wrong. 

 Since this is fractional expression with a radical in the denominator, we need to remember the bottom can't be zero and just set that denominator to equal .  Square both sides to get .  This actually means  is not acceptable but any values greater than that is good. This is the correct answer.

Range is the  value generated from a real  value.  is a linear function therefore any  values will always generate real values. Answer is all real numbers.

Range is the  value generated from a real  value. In a quadratic equation, the graph is a parabola with the graph being symmetric. Since the  is positive, we can determine the smallest  value. By determining the vertex of the graph, this will help us with the answer. The vertex of a quadratic equation is  in which  represent the values of a quadratic equation in  form.

 ,  Now let's plug into the vertex equation.

 This is only the  coordinate. We need to plug back into the quadratic equation to get the  value.

, . This means that the lowest range value of that quadratic equation is .

Answer is 

Range is the  value generated from a real  value. In an absolute value equation, most of the  values have to be at least greater than or equal to zero as all of the negative answers become positive. The only time the  value is not zero is if the graph has been shifted up or down in which we would see a numerical value outside the function. Since there is a  this means the graph has shifted down  units on the  axis. Therefore our answer is then .

Range is the  value generated from a real  value. In an absolute value equation, most of the  values have to be at least greater than or equal to zero as all of the negative answers become positive. The only time the  value is not zero is if the graph has been shifted up or down in which we would see a numerical value outside the function. Since there is none, our answer is then .

Range is the  value generated from a real  value. We know square roots have to generate all values greater than or equal to zero. However, there is a fraction inside the radical, so therefore we need to keep in mind the denominator can't equal . We know  and that as  approaches , the denominator of the fraction gets extremely small but the fractional value is extremely large therefore approaching infinity. However when  gets larger, the denominator of the fraction gets big but the fractional value is extremely small and therefore approaching zero BUT NEVER REACHES ZERO. Therefore our final answer for the range is .

Since the range is , we need to see which equation will satisfy this conditions. Based on inspection,  is not the correct answer. Since that's a linear function, all the  values will be real. Also,  is not the correct answer because we know the range of square roots are from zero inclusive and all positive numbers especially in this case.  is only indicating that no matter what value  takes on  is only . We are interested in numbers greater than or equal to six. This leaves us with the quadratic and absolute value functions only.

 In a quadratic equation, the graph is a parabola with the graph being symmetric. Since the  is positive, we can determine the smallest  value. By determining the vertex of the graph, this will help us with the answer. The vertex of a quadratic equation is  in which  represent the values of a quadratic equation in  form.

 ,  Now let's plug into the vertex equation.

 This is only the  coordinate. We need to plug back into the quadratic equation to get the  value.

, . This means that the lowest range value of that quadratic equation is . This doesn't match our expected range so this answer is wrong.

 In an absolute value equation, most of the  values have to be at least greater than or equal to zero as all of the negative answers become positive. The only time the  value is not zero is if the graph has been shifted up or down in which we would see a numerical value outside the function. Because there is a six added to the function, this means the smallest  value of this function is . This is the correct answer.

Since the range is , we need to see which functions satisfy this condition. We know for a fact  is not correct because all linear functions generate all real values as the range.  is not correct because all radicals without fractional functions inside have ranges of .  is not correct because in this absolute value equation, there is a numerical value outside the function. There is a which means the graph has shifted up  units on the  axis. Therefore this range is then  which is not correct. Lastly, the difference between  and  is in the sign in front of the square function. The shape of a quadratic equation is a U shapes ONLY WHEN THE QUADRATIC IS POSITIVE. If the quadratic equation is negative, the shape is an upside down U. This means that wherever the vertex is will become the maximum  value. We have  and  is the maximum   value so therefore the correct answer is .