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).

Domain of Definition of a Function - dCode

Tag(s) : Functions

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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.

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.

$ \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 étoile) is the reals domain excluding the value 0, also written$ ]-\infty; 0[ \cup ]0;+\infty [ $

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

$ \mathbb{R_-^*} $ (R étoile 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 [ $

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.

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

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