How to Find the Specified Set of Functions

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The defining set of a function is the set of values that make the function meaningful. In other words, this is a set of x-values that you can substitute into any given equation and get the corresponding y-value. If x is considered as the input value, then y is the output value, and the set y is called the value set (or range) of the function. To find the definite set of a function in various cases, follow the steps below.

Table of Contents

Basic knowledge

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You need to understand the concept of deterministic sets. A deterministic set is a set of input values for a function to produce an output value. In other words, the set is all the values of x that, when substituted into the function, computes the corresponding y value.

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Learn how to find the definite set of many functions. The type of function will help us to know the fastest way to find the set. Here are the basic knowledge about the types of functions that you need to know:

Polynomial functions have no roots or variables in the denominator. With a function of this form, the set is all real numbers.
Fractional function with variables in the sample. To find the defining set of this form of function, set the denominator to 0 and then exclude the x value that you find when solving the equation.
The radical function with the variable in the radical sign. To find the defined set of this form of function, simply put the term in the radical sign >0 and solve to find the x-values.
The natural logarithm (ln). Put the terms in brackets >0 and solve to find x.
Graph. The graph will show the x values in it.
The relationship between x and y. This is a list of x and y coordinate pairs. The definition set is also the list of x coordinates.

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Accurately conclude the definite set. The notation of the deterministic set is easy to learn, but it is important that you write it correctly in order to conclude the correct answer and get the maximum score when doing exercises and tests. Here are a few things to know about writing the definition set of a function:

The format when writing a specifier is an opening square/parenthesis, the beginning and end values of the domain separated by a semicolon, followed by a closing square/parenthesis. ^{[1] X Research Source}
- For example: [-1; 5). This deterministic set goes from -1 to before 5.
Square brackets [ and ] indicate that some number is included in the specified set.
- So in the example [-1; 5), the definition set includes -1.
Parentheses ( and ) indicate that the number is not included in the specified set.
- So in the example [-1; 5), 5 is not in the definite set. The deterministic set will stop arbitrarily just before 5, for example at 4,999…
The symbol “U” (meaning “combination”) is used to connect the parts of a given set that are separated by space.’
- For example: [-1; 5) U (5; 10). This definition set includes the values from -1 to 10, except for the space at 5. This could be the definition set of some function with the denominator “x – 5” “.
- If the deterministic set has a lot of space, you can use the “U” notation whenever necessary.
Use positive and negative infinity signs to represent that the definite set extends to infinity in some direction.
- Always use the ampersand ( ) with the infinity symbol (not [ ]).
Note: These formats and symbols may vary depending on where you live.
- The rules in the article are applied in Vietnam.
- Some other countries use arrows instead of infinity to show that the definite set continues to infinity in some direction.
- The spelling of the specified set in brackets also varies by region. In the United Kingdom and the United States, the beginning and the end of the specified set are separated by commas.

Find the defining set of the fractional function

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Rewrite the topic. For example, find the set of the following function:

f(x) = 2x/(x ² – 4)

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For fractional functions with variables under the denominator, set the denominator equal to 0. Since division by zero cannot be performed, when finding the determinant set of the fractional function, you need to exclude all x values that make the denominator the denominator. formula is 0. Proceed to rewrite the denominator as an equation of 0. ^{[2] X Research Source} In the above example, we have:

f(x) = 2x/(x ² – 4)
x ² – 4 = 0
(x – 2 )(x + 2) = 0
x (2; – 2)

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The conclusion of the set is determined as follows:

x is all real numbers except 2 and -2 or x ∈ R {-2; 2}

Find the definition set of the square root function

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Rewrite the topic. For example, find the set of the following function: Y = (x-7)

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Since no negative number has a real square root, the term in the radical sign must be greater than or equal to 0. Rewrite the expression in the radical sign as an equation greater than or equal to 0. ^{[3] X Research Source} How This applies to all even square roots, not just square roots. Note: negative real numbers still have odd square roots, so you can’t apply this way. Continuing the above example, we have:

x-7 0

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Isolate the variable. To find x, simply add 7 to both sides to isolate the variable x on the left side of the equation: ^{[4] X Research Source}

x-7+7 7 or x 7

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Accurately conclude the definite set. So the defining set of the function is:

D = [7;+∞)

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Find the defining set of a square root function with many solutions. Example function: Y = 1/√( ̅x ² -4). To find the definite set, square the denominator and set the equation to 0. After solving the equation, we have x ≠ (2; – 2).

Tests the range -2 and earlier (such as replacing -3) to see if this deterministic set can make the denominator greater than zero. Substituting -3 in the denominator, we have:
- (-3) ² – 4 = 5 > 0. So the set (-∞; -2) satisfies the condition.
Continue testing the range from -2 to 2. You can choose x = 0.
- 0 ² – 4 = -4, so the range -2 to 2 does not make the denominator greater than 0.
Now, test with numbers greater than 2 (such as 3).
- 3 ² – 4 = 5 > 0. So the set (2; +∞) satisfies the condition.
Conclusion of the definite set. So the defining set of the function is:
- D = (-∞; -2) U (2; +∞)

Find the definite set of the natural logarithmic function

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Rewrite the topic. For example, find the defining set of the function:

f(x) = ln(x-8)

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Write an equation with the expression in parentheses greater than 0. Since the log function is defined for all positive numbers ^{[5] X Research Source} , you need to rewrite the expression in parentheses to be an equation greater than 0 In the above example, we have:

x – 8 > 0

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Solve the equation. Now, you just need to find x by adding 8 to both sides of the equation to isolate the variable as follows ^{[6] X Research Source} :

x – 8 + 8 > 0 + 8
x > 8

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Conclusion of the definite set. The defining set of this equation is all the numbers from 8 to positive infinity ^{[7] X Research Source} :

D = (8; +∞)

Find the definite set of the function through the graph

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Look at the graph.

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Determine the x values in the graph. ^{[8] X Research Resources} Here are some tips for you:

Line graph. If the graph is in the form of a slanted line extending endlessly in either direction, then all x values will lie in it, so the definition set of this case is all real numbers.
Regular parabp plot. If the graph is a parabp with an upward or downward concave, the defining set of the function will be all real numbers because, after all, all x-coordinates will be covered.
Horizontal parabp plot. If the graph has a horizontal parabp with vertices (4,0) and extends to the right to infinity, the definition set of the function will be D = [4;∞)

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Conclusion of the definite set. Just conclude the definite set based on the type of graph you have defined. If you have a straight line equation and want to double check it, put the x-coordinate into the function and figure it out. ^{[9] X Research Source}

Find the definite set of the function through coordinate pairs

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Write down the coordinate pairs. This is simply the set of pairs of numbers given. For example: {(1; 3); (2; 4); (5; 7)}

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Write down the x coordinates. In the above example, we have x values of 1, 2 and 5. ^{[10] X Research Source}

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Conclusion of the definite set. So the set is defined as D = {1; 2; 5}

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Check to make sure that the given pairs of numbers are a function. To satisfy the condition that it is a function, every time you replace an x coordinate you will get a corresponding y coordinate. If you replace x = 3, we get y = 6, and so on. So the coordinate pairs {(1; 3); (2; 4); (5; 7)} is not a function because with coordinates x = 1, there are two values of y, respectively 4 and 5. ^{[11] X Research source}

Steps