Domain vs Range
A mathematical function is a relationship between two sets of variables. One is independent called domain and other is dependent called range. In other words, for two dimensional Cartesian coordinate system or XY system, the variable along x-axis is called as Domain and along y-axis is called as Range.
Mathematically, consider a simple relation as {(2, 3), (1, 3), (4, 3)}
In this example, Domain is {2, 1, 4}, while Range is {3}
Domain
Domain is the set of all possible input values is any relation. It means the output value in a function depends upon each member of domain. Domain value vary in different mathematical problems and depends upon the function for which is it solved. If we talk about cosine, then domain is the set of all possible real numbers either above the 0 value or below the 0 value, it could also be 0. While for square root, the domain value could not be less than 0, it should be minimum 0 or above 0. In other words, you can say that domain of square root is always 0 or positive value. For complex and real equations, the domain value is a subset of complex or real vector space. If we want to solve a partially differential equation for finding the value of domain, then your answer should lie within three dimensional space of Euclidean geometry.
For Example
If y=1/1-x, then its domain value calculated as
1-x=0
And x= 1, Hence its domain could be set of all real numbers except 1.
Range
Range is the set of all possible output values in a function. Range values are also called dependent values, because these values could only be calculated by putting the domain value in the function. In simple words, you can say that if domain value of a function y=f(x) is x, then its range value will be y.
For Example
If Y=1/1-x, then its range value will be a set of real numbers, because the values of y for every x are again real numbers.
Comparison
• The domain value is an independent variable, while range value depends upon domain value, so it is dependent variable.
• The domain is a set of all input values. On the other hand, range is a set of those output values, which a function produces by entering the value of domain.
• Here is a best theoretical example to understand the difference between domain and range. Consider the hours of sunlight during whole day. The domain is the number of hours between sunrise and sun set. While, the value of range is between 0 to maximum elevation of sun. To consider this example, you should keep in mind the hours of daylight, which vary according to season means either winter or summer. There is another thing to pay attention which is latitude. You should calculate domain and range for specific latitude.
Conclusion
No doubt, both domain and range are mathematical variables and correlate with each other, as value of range depends upon the value of domain. However, both variables have different properties and have individual identity in any one mathematical function.