Domain of Definition of a Function

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 $.

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.

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.

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 [ $

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.

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)