How To Find The Domain Of A Log Function Fraction

Mathematical concept

A function $f$ from $X$ to $Y$ . The fix of points in the red oval $10$ is the domain of $f$ .

Graph of the real-valued foursquare root function, f(x) = √ ten , whose domain consists of all nonnegative real numbers

In mathematics, the domain of a function is the gear up of inputs accepted by the role. It is sometimes denoted by $\operatorname {dom} (f)$ , where $f$ is the role.

More than precisely, given a role $f\colon X\to Y$ , the domain of $f$ is $X$ . Note that in modern mathematical language, the domain is function of the definition of a function rather than a holding of information technology.

In the special case that $Ten$ and $Y$ are both subsets of $\mathbb {R}$ , the function $f$ tin be graphed in the Cartesian coordinate system. In this case, the domain is represented on the $x$ -centrality of the graph, as the projection of the graph of the function onto the $10$ -centrality.

For a role $f\colon 10\to Y$ , the set $Y$ is called the codomain, and the ready of values attained by the role (which is a subset of $Y$ ) is called its range or image.

Any function can be restricted to a subset of its domain. The restriction of $f\colon X\to Y$ to $A$ , where $A\subseteq X$ , is written as $\left.f\right|_{A}\colon A\to Y$ .

Natural domain [edit]

If a real function $f$ is given by a formula, information technology may exist non divers for some values of the variable. In this instance, it is a partial function, and the set of real numbers on which the formula can be evaluated to a existent number is called the natural domain or domain of definition of $f$ . In many contexts, a partial role is called simply a function, and its natural domain is called just its domain.

Examples [edit]

The role $f$ defined by $f(x)={\frac {1}{x}}$ cannot be evaluated at 0. Therefore the natural domain of $f$ is $\mathbb {R} \setminus \{0\}$ .
In dissimilarity, if $f$ is the piecewise function $f(ten)={\begin{cases}1/10&10\non =0\\0&x=0\end{cases}},$ then $f$ is defined for all real numbers, and its natural domain is $\mathbb {R}$ .
The function $ten\mapsto {\sqrt {ten}}$ has equally its natural domain the non-negative existent numbers, which can be denoted by $\mathbb {R} _{\geq 0}$ , by the interval $(0,\infty )$ , or past $\{ten\in \mathbb {R} :ten\geq 0\}$ .
The tangent function $\tan x$ has every bit its natural domain the ready of all real numbers which are not of the grade ${\tfrac {\pi }{2}}+k\pi ,$ where $k$ is any integer.

Other uses [edit]

The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a continued open set.^[ane] In existent and complex analysis, a domain is an open continued subset of a real or circuitous vector space. In the report of partial differential equations, a domain is the open continued subset of the Euclidean infinite $\mathbb {R} ^{north}$ where a problem is posed (i.e., where the unknown function(s) are defined).

Set theoretical notions [edit]

For example, it is sometimes convenient in prepare theory to allow the domain of a function to be a proper class $X$ , in which example there is formally no such affair as a triple $(Ten, Y, 1000)$ . With such a definition, functions exercise non accept a domain, although some authors all the same use information technology informally afterwards introducing a role in the class $f : X \to Y$ .^[2]

Notes [edit]

^ Weisstein, Eric W. "Domain". mathworld.wolfram.com . Retrieved 2020-08-28 .
^ Eccles 1997 harvnb fault: no target: CITEREFEccles1997 (help), p. 91 (quote one, quote 2); Mac Lane 1998 harvnb fault: no target: CITEREFMac_Lane1998 (aid), p. 8; Mac Lane, in Scott & Jech 1967 harvnb error: no target: CITEREFScottJech1967 (help), p. 232; Sharma 2004 harvnb fault: no target: CITEREFSharma2004 (assistance), p. 91; Stewart & Tall 1977 harvnb error: no target: CITEREFStewartTall1977 (help), p. 89