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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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).

To calculate the definition set of a function in \( \mathbb{R} = ]-\infty ; +\infty [ \) look at the values for which the function is not defined. I.e, the values of \( x \) such as (\( f(x) \) do not exists.

'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 \( x \ge 0 \)

- 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 simply no value.

\( \mathbb{R} \) is the domain of real numbers

\( \mathbb{R^+} \) (R plus) is the domain of positive real numbers (0 included)

\( \mathbb{R^-} \) (R moins) is the domain of negative real numbers (0 included)

\( \mathbb{R^*} \) (R étoile) is the reals domain excluding the value 0.

\( \mathbb{R_+^*} \) (R étoile plus) is the domain of positive real numbers (0 exclus)

\( \mathbb{R_-^*} \) (R étoile moins) is the domain of negative real numbers (0 exclus)

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.

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Source : https://www.dcode.fr/domain-definition-function

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