The domain of a function is nothing but a set of functions is that is part of the concerned function. They are function if over the entire domain and the domain is thus the domain in the definition. Thus for a partial domain of the function, the domain of a partial function is actually a subset of the concerned domain.
A domain is a part of a function hence it can mean that for a function where it has X, Y where X is the domain and Y is the codomain. However, we cannot say a Domain is a part of a function IF the function which is defined is only represented by a graph. Sometimes the domain will proper class and there is no such thing as the graph. However, some functions will not have a domain.
For example, let us consider the domain of a cosine which is nothing but a set of numbers that are real. The domain of the square root will have numbers that are usually greater than zero or at least it will be equal to zero.
For a given function F if we determine the subset of real numbers and the function is shown in a coordinate system then we can safely assume that the domain is given in the axis which is X.
What is Natural Domain:
When we say Natural domain for any function of relation it means the maximum number of value sets for which we are defining the function or the relation. We do this with real numbers but also sometimes with complex numbers and integers. When we are considering any natural domain thus it means the numbers of sets in it are possible and it will be included in the function which is called a RANGE. When we have complex variables the function denoted is holomorphic and the domain cannot connect to the domain outside of it including the boundary. Thus D is a natural domain.
What is RANGE for a function?
Here in the field of Mathematics, A range of a function is the two related concepts:
- The function codomain
- Function image
When we talk of Range of a function there are some definitions we need to learn along with Range and domain of a function.
For one, we need to learn of Injections, Surjections and Bijections. These are function classes and they are usually defined by arguments and images. First is the Argument which is domain input and next is Images which is the output expression. These are related and mapped to one another.
Injection– This function is injective if each of the element of the codomain is mapped to the domain element. At least in the terms of equality the distinct elements of the domain will map to distinct element in the codomain. We can call injective function also as injection.
Surjection: We can call a function as a surjection if each of the element of the codomain is thus mapped to at least one element of the domain sphere. We can call the Surjective function also as Surjection.
Lastly Bijection is when in the function there is each element in the codomain which is mapped to exactly one element in the Domain. This is thus both injective and surjective. This is also known as Bijection.
How to find the Range of a relation-
First we write down its formula and that is basically the format for a function of a parabola. Let us then find the vertex of the function. no need to do anything if we are working with odd numbers and also if the function is a straight line. However, if we are working with a parabola then we know that we will have to plot the vertex.
Then we find few points in the function. To get this we need to find few X coordinates for the same and thus we can start looking for the Range of the function. Now since this is a Parabola and the coordinates are all positive so we know the range will be pointing in the upward direction. Lastly, we find the range on the Graph. Here we take the Y coordinates on the graph and also find the lowest point at the point where the graph will touch the Y coordinate. The smallest and low y coordinate is at the Vertex and the graph will here extend.
Learn more concepts on Relations From Class 12 Maths