# GRE Subject Test: Math : Domain & Range

## Example Questions

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### Example Question #1 : Calculus

What is the domain of ?

Explanation:

Step 1: We need to determine what kind of function we have here. We have a rational function.

Step 2: Since we have a rational function, the denominator cannot be equal to . We will equal the denominator to  and find the values of  that make the bottom zero.

Square both sides:

Add  to both sides of the equal sign:

Simplify:

Take the square root of both sides

However, we said earlier that these two solutions cannot be values of x, so we must change the sign:

Step 3: We need to write the solution in interval notation form.

The smallest number we can plug in for  is , and the biggest is  cannot be  and , but it can be anything else. So, we should have three intervals:

1) Between  and , which is written as
2) Between  and , which is written as
3) Between  and , which is written as

The full solution in interval notation is

### Example Question #2 : Calculus

What is the domain of:

Explanation:

Step 1: We have a square root here, where the inside of the radical must be greater or equal than , so we get a real answer.

Step 2: Set the inside equation greater or equal than zero...

Step 3: Solve for x...

... Write this in interval notation.  is the leftmost boundary, infinity is the right boundary.  has a bracket with it, infinity does not..

### Example Question #3 : Calculus

What is the domain and the range for this set of numbers?

domain is  and the range is

domain is  and the range is

domain is  and the range is

domain is  and the range is

domain is  and the range is

Explanation:

The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates).

Given this set of ordered pairs , the

domain is  and the range is

Note: If a value of a coordinate is repeated, it only gets listed once in the set.

### Example Question #4 : Calculus

The surface area of a cube is the total outside area of that cube.  A cube has a side length of at least  The surface area is a function of side length.  What is the domain (side length) and the range (surface area)?

;

;

;

;

;

Explanation:

The surface area of a cube is the total outside area of that cube. The formula to find the Area of a cube is .

A cube has a side length of at least  inches. The surface area is a function of side length. What is the domain (side length) and the range (surface area)?

The independent variable is the length of the side.  The domain consists of numbers that represent the length of sides.  The problem states that the length of the sides is at least  inches. So, the domain is all numbers greater than or equal to four.

The range consists of the numbers that corresponds with the chosen values in a function.

To find the range, find the Area of the cube with the length of the side measuring  inches; this is the smallest value that  (or the side length measurement), can be equal to.

### Example Question #1 : Domain & Range

What is the domain of the function  ?

the domain is all

the domain is all

the domain is all

the domain is all

the domain is all

Explanation:

For a function  defined by an expression with variable    the domain of  is the set of all real numbers that the variable   can take such that the expression defining the function is real.

The expression defining function contains a square root.

.

The domain is the set of all first elements of ordered pairs or

Because there cannot be a negative inside the radical sign, set the inside of the radical    and solve.   The expression under the radical has to satisfy the condition   for the function to take real values.

Solve:

Subtract  from both sides of the equation.

Divide both sides of the equation by .

Because you are dividing both sides of the equation by a negative integer  this will change the sign from     to  .

The domain or d is all

### Example Question #6 : Calculus

What is the range of the function    if the domain is  ?

Explanation:

The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates).

The domain, or all values of   .

In order to get the range or all values of    insert the given values of   (the domain) into the equation  and solve for  (the range).

The range for the function  with the domain of  is

.

### Example Question #7 : Calculus

What is the domain and range for this graph?

The domain is ; the range is .

The domain is  ; the range is .

The domain is ; the range is .

The domain is ; the range is .

The domain is ; the range is .

Explanation:

The ordered pairs represented on the graph are:

The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates).

The domain is ; the range is .

### Example Question #8 : Calculus

Find the domain for the function  ?

the domain is all

the domain is all

the domain is all

the domain is all

the domain is all

Explanation:

For a function  defined by an expression with variable  the domain of  is the set of all real numbers that the variable   can take such that the expression defining the function is real.

The expression defining function  contains a square root.       The expression under the radical has to satisfy the condition   for the function to take real values.

Solve

Add  to both sides of the equation.

Divide both sides by .

The domain is all

### Example Question #9 : Calculus

Find the domain and the range of the inverse of a relation with this set of coordinates:

The domain is  and the range is .

The domain is  and the range is .

The domain is  and the range is .

The domain is  and the range is .

The domain is  and the range is .

Explanation:

To find the domain and the range of the inverse of a relation with this set of coordinates, first find the domain and range of the set of coordinates given.

The domain is all of the -coordinates and the range is all of the -coordinates. Remember, if a coordinate is the same, it is only listed once.

The domain of  is

and the range is .

However, the question is asking for the domain and the range of the inverse.  So, the values of  (the domain) will now become the values of , (the range) and the values of  (the range) will become the values of , or the domain.

The domain for the inverse is  and the range is .

### Example Question #10 : Calculus

What is the range of the function , when the domain is  ?

Explanation:

The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates).

domain or values of

Plug all of the values of  that have been given into the equation

Even though some of the values of  are repeated, they are only listed once in the set..

The range, or all values of  for the function     are

.

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