Domain And Range

How to Find Domain and Range?

In order to find the domain of a rational function, you need to set the denominator not equal to zero. Here is an example:

In the domain of f(x) = 2/(x-3), you can set x-3 ≠ 0. By solving this, you will get x≠3. So, the domain is the set of all rational numbers except 3. You can write it in the interval notation as (-∞, 3) U (3, ∞).

In order to find the range of a rational function, you need to solve the equation for x and apply it to set the denominator not equal to zero. Here is an example:

Say you are asked to find the range of y=2/(x-3). Here, you need to solve it for x first. After doing that, you get x-3 = 2/y + 3. So, its range is y ≠0 (or) in interval notation, (-∞, 0) U (0, ∞).

 How to Use Interval Notations to Specify Domain and Range?

You can write the domain and range in interval notation, which uses the values within brackets to describe a set of numbers. You need to use a square bracket in interval notation when the set includes the endpoint and a parenthesis to mark that the endpoint is either not included or is unbounded.

Say, if a person has $200 for his weekly expense. He would need to express the interval that is more than 0 and less than or equal to 200 and write (0, 100].

Before you use interval notations, it is important to review the conventions of interval notation:

• You need to write the smallest term from the interval first.
• You should write the largest term in the interval second, following a comma.
• Parentheses, ( or ), are used to denote that an endpoint is not included, called exclusive.
• Brackets, [ or ], are used to denote that an endpoint is included, called inclusive.

Key Concepts of Domain and Range

In mathematics, functions can be compared to the operations of a vending machine. When you put a certain amount of money into it, you can choose from different types of sodas. Quite similarly, for functions, when you input different numbers, you get new numbers as a result. Domain and range are the major aspects of functions.

You have the option to use quarters and one-dollar bills to buy a soda. However, the machine will not offer you any flavour of the soda for those inputs. The domain represents the inputs, such as quarters and on-dollar bills. No matter how much amount you pay, you won’t get a pizza from a soda machine. Therefore, the range is the possible outputs we can have here.

The domain and range are the components of a function. The domain is the set of all the input values of a function, while the range is the possible output given by the function.

Domain→ Function →Range

If there is a function f: A →B such that each element of A is mapped to elements in B. Then A is the domain and B is the co-domain.

Examples of Domain and Range

• Domain and Range of Exponential Functions:

The domain of exponential functions is always equal to all real numbers since there are no restrictions on the values that x can take. On the other hand, the range of exponential functions is equal to the values above or below the horizontal asymptote.

Here is an example:

An exponential function as simple as f (x) = 2^x has a domain equal to all real numbers. But its range is equal to only positive numbers, where y > 0. The function f (x) never takes a negative value.

You also need to consider the fact that the function never reaches the value of 0 even though it gets very close as x approaches negative infinity.

Domain and Range of Trigonometric Functions:

The domain and range of trigonometric functions are the input values and the output values of trigonometric functions, respectively. The trigonometric functions’ domain indicates the values of angles where the trigonometric functions are defined, and the range of trigonometric functions gives the resultant value of the trigonometric function corresponding to a particular angle in the domain.

Here is an example:

Say you are asked to determine the domain and range of y = 3 tan x.

Solution: We know that the domain and range of trigonometric function tan x is given by, Domain = R - (2n + 1)π/2, Range = (-∞, +∞)

It is to be noted that the domain is given by the values that x can take. Thus the domains of tan x and 3 tan x are the same. Hence the domain of y = 3 tan x is

R - (2n + 1)π/2

The range of tan x is (-∞, +∞) ⇒ -∞ < y < ∞

⇒ -∞ < tan x < ∞

⇒ -∞ < 3 tan x < ∞ [As multiplication of ∞ by 3 results in ∞ only]

• Domain and Range of a Quadratic Function

The domain and range of a quadratic function y=a(x-h)2+k determine the nature of the parabola. Depending on the value of the variable, the hyperbola can be upwards or downwards or facing to the left or to the right.

• y ≥ k, if the function has a minimum value, that is when a>0 (parabola opens up)
• y ≤ k, if the function has a maximum value, that is when a<0 (parabola opens down)

How to Find the Domain and Range of a Relation?

Before you learn about the way to find the domain and range of a relation, you need to understand what relations are. A relation is a rule that relates a certain element from one set to the other set. The domain is the set of all first elements of the ordered pairs. On the other hand, the range is the set of all second elements of the ordered pairs.

However, range generally includes only one of those elements used by the function. There is a trick in the range, i.e. set B can be equal to the range of relation or bigger than that. It is because there can be elements in set B that may not be related to set A.