The domain of a function is the set of numbers that can be entered in the function itself. In other words, it is the set of Xs that you can put in a certain equation. The set of possible Y values is called the range or rank of the function. If you want to learn how to find the domain of a function in different situations, just follow these steps.
Steps
Method 1 of 6: Learn the Basics
Step 1. Learn the domain definition
The domain is defined as the set of input values for which the function produces an output value. In other words, the domain is the set of values of x that can be inserted into a function to produce a value of y.
Step 2. Learn how to find the domain of different functions
The specific type will determine the best method for finding a domain. Here are the basics you need to know about each type of function, which will be explained in the following section:
- Polynomial function without radicals or variables in the denominator. For this type of function, the domain consists of all real numbers.
- Polynomial function with variables in the denominator. To find the domain of such a function, you need to exclude the X values that make the denominator equal to zero.
- Function with unknown in the radical. To find the domain of such a function, it is necessary to take the expression contained within the root, place it greater than zero and solve the inequality.
- Function with natural logarithm log (ln). We must ask the argument of the logarithm greater than zero and solve.
- Graphic. We need to look for which X intersects the horizontal axis.
- Relation. It is the list of the X and Y coordinates. The domain will simply be the list of all the X's.
Step 3. Write the domain correctly
Learning the correct domain notation is easy, but spelling it correctly is important to get the right answer and get the most out of a class test or exam. Here are some things you need to know to be able to write the domain of a function.
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The format for indicating the domain is an opening parenthesis, followed by the two ends of the domain separated by a comma, followed by a closing parenthesis.
For example, [-1, 5). This means that the domain ranges from -1 included to 5 excluded
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Use square brackets, such as [and] to indicate that the number is included in the domain.
In the example, [-1, 5), the domain includes -1
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Use "(" and ")" to indicate that a number is not included in the domain.
In the example, [-1, 5), 5 is not included in the domain. Domination stops arbitrarily just before 5, that is 4, 999 …
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Use "U" ("union") to connect parts of the domain that are separated by a range. '
- For example, [-1, 5) U (5, 10] means that the domain is from -1 to 10 inclusive, but that there is a range of 5 in the domain. This could be the result, for example, of a function with "x - 5" in the denominator.
- You can use as many "U" as you need, in the case of a domain with more than one range.
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Use the symbols of positive infinity or negative infinity to indicate that the domain goes to infinity in either direction.
With infinity symbols, always use (), not
Method 2 of 6: Finding the Domain of a Fratta Function
Step 1. Write down the problem
Suppose it is the following:
f (x) = 2x / (x2 - 4)
Step 2. In the case of a fractional function, equal the denominator to zero
To find the domain of a function with unknown in the denominator, you must exclude the values of x that make the denominator equal to zero, because it is not possible to divide by zero. So write the denominator as an equation equal to 0. Here's how:
- f (x) = 2x / (x2 - 4)
- x2 - 4 = 0
- (x - 2) (x + 2) = 0
- x ≠ (2, - 2)
Step 3. Read the domain
That's how:
x = all real numbers except 2 and -2
Method 3 of 6: Finding the Domain of a Function Under Square Root
Step 1. Write down the problem
Suppose it is: Y = √ (x-7)
Step 2. In square roots, the radicand (the expression under the root symbol) must be equal to or greater than 0
Then write the inequality so that the radicand is greater than or equal to 0. Note that this applies not only to square roots, but to all roots with even exponents. It does not hold for roots with odd exponents, because it is possible to have negative numbers under odd roots. That's how:
x-7 ≧ 0
Step 3. Isolate the variable
At this point, to bring the X to the left side of the equation, just add 7 on both sides, in order to obtain:
x ≧ 7
Step 4. Write the domain correctly
That's how:
D = [7, ∞)
Step 5. Find the domain of a square rooted function with multiple solutions
Suppose we have the following function: Y = 1 / √ (̅x2 -4). By breaking down the denominator and equating it to zero, we get x ≠ (2, - 2). Here's how to proceed:
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Now check the interval less than -2 (putting X equal to -3, for example) to see if a number less than -2 placed in the denominator gives a number greater than zero. It is true.
(-3)2 - 4 = 5
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Now try with the range between - 2 and 2. Take 0, for example.
02 - 4 = -4, so you see that numbers between -2 and 2 don't fit.
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Now try with a number greater than 2, for example +3.
32 - 4 = 5, then numbers greater than 2 are fine.
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When you are done, write the domain. It should be written like this:
D = (-∞, -2) U (2, ∞)
Method 4 of 6: Finding the Domain of a Function with a Natural Logarithm
Step 1. Write down the problem
Suppose we have:
f (x) = ln (x-8)
Step 2. Put the expression in brackets greater than zero
The natural logarithm must be a positive number, so you must put the expression greater than zero. That's how:
x - 8> 0
Step 3. Solve
Isolate the variable X and adding eight on both sides. You get:
- x - 8 + 8> 0 + 8
- x> 8
Step 4. Write the domain
Note that the domain of this equation is composed of all numbers greater than 8 up to infinity.
D = (8, ∞)
Method 5 of 6: Finding the Domain of a Function Using a Graph
Step 1. Take a look at the graph
Step 2. Check the X values that are included in the graph
It's easier said than done, but here are some tips:
- A straight line. If the graph consists of a line that extends to infinity, all Xs will be taken, so the domain includes all real numbers.
- A normal parable. If you see a parabola pointing up and down, the domain will be composed of all real numbers, because in the end all the numbers on the X axis will be covered.
- A horizontal parabola. For example, if you have a parabola with the vertex at (4, 0) extending to infinity to the right, the domain is D = [4, ∞)
Step 3. Write the domain
It depends on the type of chart you are working on. If you are uncertain, enter the X coordinates in the function to check.
Method 6 of 6: Finding the Domain of a Function with a Relation
Step 1. Write the relationship, which is made up of a series of X and Y coordinates
Suppose we work with the following coordinates: {(1, 3), (2, 4), (5, 7)}
Step 2. Write the X coordinates
They are: 1, 2, 5.
Step 3. Write the domain
D = {1, 2, 5}
Step 4. Make sure the relationship is a function
To verify this, for each value of X you should always get the same Y coordinate. For example, if X is 3, you should always get only 6 as Y and so on. The following relation is not a function because, for the same value of X, two different values of Y are obtained: {(1, 4), (3, 5), (1, 5)}.
