Domain of a function
In mathematics, the domain (set of definition or set of items) of a function f:X→ → And{displaystyle f:Xto Y} is the set of existence of itself, that is, the values for which the function is defined. It is the set of all objects that can transform, denotes Domf{displaystyle operatorname {Dom} _{f}}, Dom (f){displaystyle operatorname {Dom} (f)} or Df{displaystyle D_{f},}. In Rn{displaystyle mathbb {R} ^{n} is called domain to a related, open and non-empty set.
On the other hand, the set of all possible outcomes of a given function is called the codomain of that function.
Definition
The domain of a function f:X→ → And{displaystyle f:Xto Y} defined as the whole X{displaystyle X} of all elements x{displaystyle x} for which the function f{displaystyle f} associates and{displaystyle and} belonging to the group And{displaystyle Y} of arrival, called codomain. This, formally written: is a fusion of all values
- Df={x한 한 X:consuming consuming and한 한 And:f(x)=and!{displaystyle D_{f}={xin X:exists ;yin Y:f(x)=y}
Properties
Given two real functions:
f:: X1→ → Randg:: X2→ → R{displaystyle fcolon X_{1}to mathbb {R} ,qquad {mbox{y}quad gcolon X_{2}to mathbb {R} ,}
It has the following properties:
- D(f+g)=X1 X2{displaystyle D_{(f+g)}=X_{1}cap X_{2}}
- D(f− − g)=X1 X2{displaystyle D_{(f-g)}=X_{1}cap X_{2}}
- D(f⋅ ⋅ g)=X1 X2{displaystyle D_{(fcdot g)} =X_{1}cap X_{2}}}
- D(f/g)={x한 한 (X1 X2)日本語g(x)I was. I was. 0!{displaystyle D_{(f/g)}={xin (X_{1}cap X_{2})
Calculating the domain of a function
For the accurate calculation of the domain of a function, the concept of restriction must be introduced in the real field. These restrictions will help to identify the existence of the domain of a function. The most used are:
Logarithm of a function
Logarithms are not defined for negative numbers nor for zero, therefore any function contained within a logarithm must necessarily be strictly greater than zero. For example:
- log (x2− − 9){displaystyle log(x^{2}-9)}
For the property mentioned above, it is observed that for this function to be well defined, necessarily 0}" xmlns="http://www.w3.org/1998/Math/MathML">x2− − 9▪0{displaystyle x^{2}-90}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29694fd232d0ef8a3e03fba9658eb9a7c2a7df8d" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;"/>; clearing, two solutions are obtained 3}" xmlns="http://www.w3.org/1998/Math/MathML">x▪3{displaystyle x 2005}
3}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca13c1461fe5c28b6ba92af1e60b99cde4a53648" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;"/> and <math alttext="{displaystyle xx.− − 3{displaystyle x vis-3}<img alt="{displaystyle x. The union of both solutions represents the domain of the function, which is defined as the set (-∞, -3) U (3, +∞).
Fractions
Other properties of mathematics can help to obtain the domain of a function and exclude points where it is not defined. For example, a function that has the form of a fraction will not be defined when the denominator is equal to zero.
Examples
Some domains of real functions of real variables:
- f(x)=x2{displaystyle f(x)=x^{2},!} The domain of this function, as well as any polynomial and exponential function, is R{displaystyle mathbb {R} }.
- f(x)=1x{displaystyle f(x)={frac {1}{x}}}}} The domain of this function is R− − {0!{displaystyle mathbb {R} -lbrace 0rbrace } since the function is not defined for x = 0.
- f(x)=log (x){displaystyle f(x)=log(x),!} The domain of this function is (0,+∞ ∞ ){displaystyle (0,{+}infty)} since the logarithms are defined only for positive numbers.
- f(x)=x{displaystyle f(x)={sqrt {x}}} The domain of this function is [chuckles]0,+∞ ∞ ){displaystyle lbrack 0,{+}infty}} because the index root to a negative number does not exist in the body of the real ones.
