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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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Tag(s) : Functions

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# Domain of Definition of a Function

## Calculation of a Function's Domain of Definition

 Solving Domain Set R (Reals) C (Complex)

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).

## Answers to Questions

### 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* means?

$\mathbb{R}$ 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 $]-\infty; n[ \cup ]n;+\infty [$

### 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.

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