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Domain of Definition of a Function

Tool to calculate the domain of definition of a function f(x): the set of values x which exists through f (from the equation of the function or its curve).

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Domain of Definition of a Function -

Tag(s) : Functions

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Domain of Definition of a Function

Function's Domain Calculator





Derivative Domain Calculator

Answers to Questions (FAQ)

What is a domain of definition of a function? (Definition)

A function $ f $ in $ \mathbb{R} $, has a domain or set of definition, denoted $ D_f $, that is the set of real numbers which admit an image by the function $ f $.

Example: The definition domain for the function $ x^3 $ is $ \mathbb{R} = ] -\infty ; +\infty [ $ as every real number has a cubed value.
The definition set of the function $ \sqrt{x} $ is $ \mathbb{R^+} = [0;+\infty [ $ as only positive real numbers have a square root.

How to calculate the domain of definition of a function?

To calculate the definition set of a function in $ \mathbb{R} = ]-\infty ; +\infty [ $, look at the values for which the function exists and those for which it does not exist, ie all the values of variable $ x $ such that $ f(x) $ is not defined.

From the equation of the function

There are generally 3 main cases of undefined values (for real functions):

division by $ 0 $ (null denominator), since $ 0 $ has no inverse

— negative square root: $ \sqrt{x} $ is defined only for $ x \ge 0 $ in $ \mathbb{R} $

— negative logarithm: $ \log(x) $ is defined only for $ x > 0 $

dCode will compute and check the values without inverse by the function $ f $ and return the corresponding interval for the domain of the function.

Example: Take $ f(x) = \sqrt{1-2x} $, since a root can not be negative, calculate the values such that $ 1-2x \ge 0 \iff x \le 1/2 $. Thus $ f(x) $ exists if and only if $ x \le 1/2 $. The domain of definition can be written $ D = ] -\infty ; 1/2 ] $

From the curve of the function

It is a question of looking at the values for which the curve has no point. Either because there is a vertical asymptote, or because there is no defined value.

What does domains R+ or R- or R* mean?

In order to simplify and shorten the writing of the intervals of the domains of definition, some domains are abbreviated as follows:

$ \mathbb{R} $ is the domain of real numbers, also written $ ]-\infty ;+\infty [ $

$ \mathbb{R^+} $ (R plus) is the domain of positive real numbers (0 included), also written $ [0;+\infty [ $

$ \mathbb{R^-} $ (R moins) is the domain of negative real numbers (0 included), also written $ ]-\infty; 0] $

$ \mathbb{R^*} $ (R asterisk) is the reals domain excluding the value 0, also written$ ]-\infty; 0[ \cup ]0;+\infty [ $

$ \mathbb{R_+^*} $ (R asterisk plus) is the domain of positive real numbers (0 exclus), also written $ ]0;+\infty [ $

$ \mathbb{R_-^*} $ (R asterisk moins) is the domain of negative real numbers (0 exclus), also written $ ]-\infty; 0[ $

$ \mathbb{R}\backslash\lbrace{n}\rbrace $ is the domain of real numbers but without the number $ n $, also written $ ]-\infty; n[ \cup ]n;+\infty [ $

What is an antecedent?

If a function y = f (x) then the number y is called the image of x, and x is called an antecedent of y with the function f in the definition domain D.

What is the existence domain of a function?

The domain of existence and the domain of definition of a function are identical, it is the same concept.

What is the difference between a domain of definition and a domain of validity?

Sometimes these two terms are misused to describe the same thing, however, they theoretically have slightly different meanings.

The domain of definition is the set of values ​​for which the function is mathematically defined.

The domain of validity is the set of values ​​for which the function, or an approximation, is correct in a given context. (And sometimes this domain of validity is equal to the domain of definition)

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Domain of Definition of a Function on dCode.fr [online website], retrieved on 2024-12-03, https://www.dcode.fr/domain-definition-function

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