The domain of a function is the group of numbers that fits in a given function. In other words, it's the group of x values that you can put into an equation. The group of possible y values is called the function range. To know how to calculate the domain of a function in different situations, just follow the steps below.
Steps
Method 1 of 6: Learning the Basics
Step 1. Learn the domain definition
Before you can begin to find domain specific functions, you need to first have a strong understanding of what a domain actually is. The domain is defined as a series of input values for which the function produces an output value. In other words, the domain is the complete value of xvalues that can be used in a function to produce yvalues.
Step 2. Learn how to find mastery of a variety of roles
The function type will determine which method is best to use. Below are the basic topics you need to know about each role, which will be explained in the next agenda:

A polynomial function with no radicals or variables in the denominator.
For this type of function, the domain consists of all real numbers.

A function with a fraction with a variable in the denominator.
To find the domain of this type of function, leave the bottom equal to zero and exclude the value of x you find when solving the equation.
 A function with a variable inside a radical symbol.' To find the domain of this type of function, just leave the terms inside the stem symbol at >0 and solve the problem to find the proper values for x.

A function using the natural logarithm ln(x).
Just leave the terms in parentheses at >0 and solve the problem.

A graph.
Use the graph to see which values are suitable for x.

A relationship.
This will be a list of x and y coordinates. Your domain will simply be a list of x coordinates.
Step 3. Correctly determine the domain
Correct mathematical representation of a domain is relatively easy, but it is important to write it correctly to express the correct answer and get more points on academic exams. Here are some tips for writing the domain of a function:
 The format for expressing the domain is an open parenthesis/bracket followed by 2 domain endpoints separated by a comma, followed by closed parenthesis/brackets.
For example, [1, 5). That means the domain goes from 1 to 5
 Use square brackets such as [and] to indicate that a number is included in the domain.
Returning to our example, [1, 5), the domain includes 1
 Use parentheses such as (e) to indicate that a number is not included in the domain.
So, in the example, [1, 5), 5 is not included in the domain. The domain must stop before 5, for example on 4999…
 Use “U” (which stands for “union”) to link the parts of the domain that are separated by a space.'
 For example, [1, 5) U (5, 10] This means that the domain goes from 1 to 10, but there is a space in the domain at 5. This could be the result of a function with “x  5” in the denominator.
 You can use the "U" symbol as needed if the domain contains multiple spaces.
 Use the infinity and negative infinity symbols to show that the domain extends infinitely in one direction.
Always use (), not [], with infinity symbols
Method 2 of 6: Finding the Domain of a Function with a Fraction
Step 1. Write the problem
Suppose you have to solve the following problem:
 f(x) = 2x/(x^{2}  4)
Step 2. For fractions with a variable in the denominator, leave the denominator equal to zero
When calculating the domain of a function with a fraction, you must exclude all values of x that leave the denominator equal to zero, as it is impossible to divide a number by zero. Then write the denominator as an equation and leave it equal to zero. See how:
 f(x) = 2x/(x^{2}  4).
 x^{2}  4 = 0.
 (x  2)(x + 2) = 0.
 x ≠ (2,  2).
Step 3. Define the domain
See how:
x = all real numbers except 2 and 2
Method 3 of 6: Finding the Domain of a Function with a Square Root
Step 1. Write the problem
Imagine solving the following problem: Y =√(x7)
Step 2. Leave the terms inside the radicand so that they are greater than or equal to zero
Since you can't get the square root of a negative number, you can get the square root of zero. Therefore, leave the terms inside the radicand so that they are greater than or equal to zero. Remember that this goes not only for square roots, but also for all even number roots. However, this is not true for oddnumbered roots, as it is perfectly acceptable to have negative numbers in oddnumbered roots. Watch:
x7 ≧ 0
Step 3. Isolate the variable
Now isolate x on the left side of the equation and add 7 on both sides to get the following result:
x ≧ 7
Step 4. Define the domain
See how:
D = [7, ∞)
Step 5. Find the domain of a function with a square root when there are multiple solutions
Suppose you are working with the following function: Y = 1/√(̅x^{2} 4). By factoring the denominator and leaving it equal to zero, you get x ≠ (2,  2). Check out the breakdown:
 Now check the area below 2 (when fitting 3, for example) to see if numbers below 2 can be fitted into the denominator to result in a number greater than 0.
 (3)^{2}  4 = 5
 Now check the area between 2 and 2. Let's choose 0, for example.
 0^{2}  4 = 4, so you see that numbers between 2 and 2 won't do.
 Now try a number above 2, like +3.
 3^{2}  4 = 5, so numbers above 2 are valid.
 Finally, write the domain. Here's the template:
D = (∞, 2) U (2, ∞)
Method 4 of 6: Finding the Domain of a Function Using a Natural Algorithm
Step 1. Write the problem
Suppose you are working with the following problem:
f(x) = ln(x8)
Step 2. Leave terms inside parentheses greater than zero
The natural algorithm has a positive number, so the terms inside the parentheses are greater than zero for this to be possible. Watch:
x  8 > 0
Step 3. Solve the problem
Isolate variable x by adding 8 on both sides. Note:
 x  8 + 8 > 0 + 8
 x > 8
Step 4. Define the domain
Show that the domain for this equation is equal to all numbers greater than 8 to infinity. See how:
D = (8, ∞)
Method 5 of 6: Finding the Domain of a Function Using a Graph
Step 1. Look at the chart
Step 2. Pay attention to the x values included in it
Sounds easy, but here are some caveats:
 A line. If you see a line on the graph that extends to infinity, it means that all versions of x are valid because the domain consists of all real numbers.
 A normal parable. If you find a parabola facing up or down, then the domain will be made up of all real numbers, as all numbers on the xaxis will be valid.
 A side parable. If you see a parabola with a vertex at (4, 0) that extends infinitely to the right, then its domain is D = [4, ∞)
Step 3. Define the domain
Define the domain based on the chart you are working with. When in doubt, but knowing the equation on the line, fit the x coordinates back to the function to verify that the result is correct.
Method 6 of 6: Finding the Domain of a Function Using a Relation
Step 1. Write down the relationship
A relationship is nothing more than a list of x and y coordinates. Imagine working with the following coordinates: {(1, 3), (2, 4), (5, 7)}
Step 2. Write the x coordinates
They are: 1, 2, 5.
Step 3. Define the domain
D = {1, 2, 5}.
Step 4. Check if the relationship is a function
For a relation to be a function, every time you put in a numeric x coordinate, you must get the same y coordinate. So if you put 3 for x, you should always get 6 for y, and so on. The following relation is not a function because in it you get two different values for "y" for each value of "x": {(1, 4), (3, 5), (1, 5)}.