Domain Codomain and Range Functions

Let us take and daily use the example of the operations of a vending (soda) machine that can be related to mathematical functions. You can choose from a variety of sodas after depositing a particular amount of money. Similarly, we input distinct numbers into functions and get new numbers as a consequence. The domain and range of functions define them. A drink can be purchased using quarters or one-dollar bills. The machine will not give you any soda flavour if you put in cents. As a result, the domain depicts the possible inputs, which are quarters and one-dollar bills. You won’t receive a cheeseburger from a soda machine, no matter how much you pay. Let’s look at how to find the domain and range of a function, as well as how to graph them. 

What is Domain, Codomain & Range?

A domain of the given relation and range can be defined as the sets of all the x-coordinates and all the y-coordinates of ordered pairs, respectively. 

For example, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3), (5,4)}, then:

• Domain = the set of all x-coordinates = {1, 2, 3, 4, 5}
• Range = the set of all y-coordinates = {2, 3, 4}

Domain and Range of a Function:

The Domain of the given function and range are its two fundamental components. The domain of a function is the set of all its input values, whereas the range is the function’s possible output. Range is a domain function. A is the domain, and B is the co-domain, if there is a function f: A → B that maps every element of A to elements in B.

Domain of a Function:

A function’s domain refers to “all the values” that go into it. The domain of the given function can be defined as the set of all the function’s possible inputs. Consider this box as a f(x) = 2x function. The domain is simply the set of natural numbers, and the output values are referred to as the range, when the values x = {1,2,3,4,…} are entered. However, because f(x) = 2x is defined for all real values of x, its domain is the set of all real numbers indicated by (-∞, ∞). The following are the general formulas for determining the domain of various sorts of functions. R is defined as the set of all real numbers where,

• R is the domain of any polynomial function (linear, quadratic, cubic, and so on).
• A square root function’s domain is √x is x≥0.
• R is the domain of an exponential function.
• x>0 is the domain of a logarithmic function.
• Set the denominator ≠ 0 to get the domain of a rational function y = f(x).

Range of a Function:

The range of a function is the collection of all its outputs.Example: Consider the function f: A→ B, where f(x) = 2x and A and B each represent a set of natural numbers. A is the domain, while B is the co-domain in this case. The range is then the output of this function. The range is defined as a group of even natural numbers. The domain elements are referred to as pre-images, while the mapped elements of the co-domain are referred to as images. The range of the function f is the set of all representations of the domain’s elements (or) the set of all the function’s outputs. The general formulas for determining the range of different types of functions are listed below. R is defined as the set of all real numbers in this case.

• R is the range of a linear function.
• The range of a quadratic function y = a(x-h)2 + k is y≥k, if a>0 and y≤k, if a<0.
• A square root function’s range is y≥0.
• An exponential function has a range of y>0.
• R is the range of a logarithmic function.
• Solve for x and set the denominator ≠ 0 to obtain the range of a rational function y = f(x).

Codomain of a Function:

A function’s codomain, or set of destinations, is the set into which all of the function’s output is restricted to fall. In the notation f: X → Y, it is the set Y. The term range is sometimes used interchangeably to refer to a function’s codomain or image. The codomain is the set of all a function’s possible output values. It’s not the same thing as a range. The range is the collection of all values generated by applying the function to domain values. The absolute value function, for example, can be thought of as a function with domain R and codomain R. Its range, on the other hand, is the collection of all non-negative real numbers. As a result, its codomain is significantly bigger than its range.

Conclusion

A function’s domain refers to “all the values” that go into it. The domain of a function is the set of all the possible output of the given function inputs. A function’s codomain, or set of destinations, is the set into which all of the function’s output is restricted to fall.