Is the Domain of the Function Represented by the Data Shown in the Graph Discrete or Continuous

A function links an input to an output. It is similar to a machine that holds an input and an output. And the output is associated somehow with the input. There are several alternatives to think about functions, but there are always three main components:

• The input
• The relationship
• The output

A relation where every input has a particular output is the function math definition. Functions in mathematics can be correlated to the real-life operations of a printer machine. When we insert a certain amount of paper combined with some commands we obtain printed data on the papers.

Also, read about Statistics here.

Similarly, for functions, we input varying numbers and we receive new numbers as the outcome of the operation performed. The domain and range are the main characters of a function. The domain of a function is the inputs of the given function on the other hand the range signifies the possible outputs we can have.

Through this article on the domain of a function, we will aim to learn about the domain meaning in math along with questions to understand how to find the domain of a function and so on.

What is the Domain?

The domain in math can be taken as a set of the values that go inside a function; furthermore, the range implies all the values that come out.

A function relates an input to output, that is function links each element of a set with specifically one element of another set. There are various ways for the representation of functions, let take a quick overview of each of them.

Representation of Functions

The functions require to be designed to display the domain values and the range values and the relationship or link between them. There are three distinct forms of representation of functions and they are Venn diagrams, graphical forms, and roster patterns. Three of the patterns are discussed below.

Venn Diagram

The Venn diagram is a powerful form for describing the function. The Venn diagrams are normally displayed by two circles with arrows combining the components in each of the circles. The domain is shown in one circle and the range values are placed in another one. Also, the function specifies the arrows, and how the arrows relate the different elements in the two given circles.

Graphical Form

Functions are straightforward to understand if they are represented in the graphical pattern with the use of the coordinate axes. Expressing the function in the graphical form helps us to learn the changing operation of the functions if the function is progressing or declining. The domain of the function is represented on the x-axis, and the range of the function is plotted on the y-axis respectively.

Also, read about Sequences and Series here.

Roster Form

Roster notation or the roster form of a set is a simple mathematical representation of the set. The domain and range of the function are expressed in brackets with the first component of a pair denoting the domain and the second component expressing the range.

Let us try to surmise this with the help of a simple example. For a function of the pattern \(f(x) = x^{3}\), the function is represented as {(1, 1), (2, 8), (3, 27), (4, 64)}. Herein the first element denotes the domain or the x value and the second component signifies the range or the f(x) value of the function.

With the knowledge of the representation of functions let us now proceed towards the more detailed analysis of the domain in mathematics.

The domain of a Function

In mathematics, we can associate a function to a machine that creates some output in correlation to a given input. By taking an example of a coin stamping tool.

When we enter a coin into the coin stamping tool, the result is an impressed and flattened piece of metal. By viewing a function, we can correlate the coin and the flattened part of metal with the domain and range. In this example, a function is supposed to be the coin stamping machine.

Exactly like the coin stamping device, which can only offer a single flattened piece of metal at a time, a function operates in the same manner by transmitting out one result at a time.

Learn more about Relations and Functions here.

Domain and Codomain of a Function

If f: P → Q is a function, then the set P is named as the domain of the function f and set Q is designated as the co-domain of the function f.

Natural domain

The natural domain of a function denotes the maximum set of values for which the function is determined, typically in the reals but sometimes with the integers or complex numbers also.

For instance, the fundamental domain of square root is the non-negative real values when viewed as a real number function. When studying a natural domain, the set of potential values of the function is typically declared its range.

Range of a Function

The range of a function is the set of all its outputs. If f: P → Q is a function, then the range of f consists of those components of Q which are connected with at least one element of P. It is expressed by f(P).

Thus, f(P) = {y : y = f(x) for some x ∈ P}

The components of the domain are named pre-images and the components of the co-domain which are mapped are named the images. Here, the range of the function is the set of all images of the components of the domain.

How to Find the Domain of a Function?

Domain, codomain and range are special titles for what can go into the function, and what can come outside of a function:

• What can fit into a function is the functional domain definition.
• What may probably appear out of a function is termed as the codomain of a function.
• What appears out of a function is named the range of a function.

We can arrange the domain of a function either algebraically or by the graphical approach. To obtain the domain of a function algebraically, we need to solve the equation to get the values of x. However, different types of functions have their means of determining the domain.

Domain math example 1:

Suppose X = {2, 3, 4, 5,6}, f: X → Y, where R = {(x,y) : y =3x+1}.

Domain = the input values of the given function, thus domain = X = {2, 3, 4, 5,6}

Range = the output values of the given function = {7, 10, 13, 16, 19}.

Check out this article on Sets.

How to Find the Domain and Range of an Equation?

One thing that should be kept in mind while determining domains and ranges is that we need to acknowledge what is physically achievable or meaningful in real-world cases. We are also required to consider what is mathematically allowed.

For example, we cannot incorporate any input value that directs us to take an even root of a -ve number if the domain and range consist of real numbers only. Conversely, in a function expressed as a formula, we cannot add any input value in the domain that would drive us to divide by zero. Consider the below example to understand the same:

Solved Example 2:

• The set "A" in the above figure denotes the domain and the set "B" signifies the codomain.
• Furthermore, the set of components that get pointed to in B that are the original values produced by the function. These values are termed as the range which is also called the image of the function.

And we obtain:

• Domain: {1, 2, 3, 4, 5}.
• Codomain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
• Range: {2, 3, 4, 5, 6}

Solved Example 3:

Consider another simple example of a function like ᠎᠎᠎᠎᠎᠎᠎\(f(x) = x^{3}\) will have the domain of the elements that go into the function.

Then the domain of a function will have numbers {1, 2, 3,…} and the range of the given function will have numbers {1, 8, 27, 64…}.

Similar to the above function take another examples with a function \(g(x) =x^{3}\).Here we can have the domain of integers like {…,-3,-2,-1,0,1,2,3,…},for which the range is the set {…,-27,-8,-1,0,1,8,27,…}.

Check more topics of Mathematics here.

Here comes a question: does every function have a domain?

The answer would be yes, though, in more simplistic mathematics, we never see this because the domain is something assumed like all numbers that will operate. Or if we are considering whole numbers, the domain is supposed to be whole numbers, etc.

We can address the domain and range in interval notation, which accepts values within brackets to define a set of numbers. In interval notation, we apply a square bracket [] when the set involves the endpoint and a parenthesis () to show that the endpoint is either not covered or the given interval is unbounded.

Solved Example 4:

Consider the relation {(2,7),(0,6),(1,5),(3,8),(1,9),(6,10)}

Here, the relation is drawn as a set of ordered pairs. The domain is the set of x -coordinates which include the values {0, 1, 2, 3, 6}, and the range implies the set of y -coordinates, {7, 6, 5, 8, 9, 10}. Note that the domain element 1 is connected with more than one range element, (1,5) and (1,9) therefore this is not a function.

Solved Example 5:

For the given function:

The domain holds the set {A,B,C,E} . Here D is not in the domain, as the function is not specified for D.

The range is the set {3, 2, 5, 6}. 1 is not inside the range, since no alphabet in the domain gets mapped to 1.

Learn the various concepts of the Line Graph with this article.

Domain and Range from Graphs

Another approach to identifying the domain and range of functions is by applying graphs. For the given below graph, the domain points to the collection of probable input values. The domain of a graph includes all the input values displayed on the x-axis.

On the other hand, the range is the collection of possible output values presented on the y-axis. Let us understand how to find the domains of the toolkit functions.

Solved Example 6:

Let us take an example to understand how to find domain and range of a graph function:

For the given graph function; the domain is x≥−4 as x cannot be smaller than −4. To examine why, attempt some numbers less than −4 say −7 or−12 and some other values which are more than −4 like that of −3 or 6 in your calculator and check the answer.

The only ones that would work and provide us with a solution are the ones that are greater than or equivalent to −4. This will make the number under the square root a positive one.

Domain of Exponential Functions

The function \(y = a^{x}\), a ≥ 0 is determined for all real numbers. Therefore, the domain of the exponential function is the complete real line. Domain = R, Range = (0, ∞)

Consider the graph for the function f: \(2^{x}\).

Domain of Trigonometric Functions

Look at the below graph of the sine function and cosine function. We can find that the value of the functions swings between -1 and 1 and it is defined for all real numbers.

Therefore the Domain of such functions is the set R.

Also, read about Multiple Line Graphs here.

Domain of an Absolute Value Function

The absolute function say y=|ax+b| is specified for all real numbers. Therefore, the domain of the absolute value function comprises the collection of all real numbers.

Consider an example: |8-x|

|8-x| ≥ 0

8 – x ≥ 0

x ≤ 8.

For the absolute value function represented by f(x)=|x|, there exists no limitation on x values. Though, because the absolute value is determined as a distance from zero, the output can simply be greater than or equivalent to zero.

Domain of a Square Root Function

For the square root function \(f(x)=\sqrt{x}\), we cannot get the square root of a negative real number, therefore the domain must be zero or greater. If we include imaginary numbers then things can get more complex, however in most cases, we are only required to consider real numbers.

To determine the domain of a square root function we need to solve the inequality x ≥ 0 with x substituted by the radicand. Let us take an example of how you can find the domain of a function:

Solved Example 7:

\(f(x)=\sqrt{x+6}\).

• Herein sets the radicand (i.e.x + 6) equal to x in the inequality.
• This provides us with the inequality of x + 6 ≥0.
• This you can solve by deducting 6 by both sides which further provides you with a solution of x ≥ −6.
• This implies that the domain holds all values of x greater than or equal to −6.
• We can also write this as [ −6, ∞). where the bracket on the left shows that −6 is a particular limit while the brackets on the right show that ∞ is not. As the radicand cannot be -ve value, we are only required to calculate for positive or zero values.

Read more about Limits and Continuity here.

Domain of a Constant Function

For the constant function represented by f(x)=c, the domain consists of solely real numbers; this implies that there are no limitations on the input. Consider the below graph for y=2.

Domain of Identity Function

For the identity function represented by f(x)=x, there is again no restriction on the value of x. Both the domain and range are the collection of all real numbers. Check out the below graph.

Also, read about x-axis and y-axis here.

Domain of Quadratic Functions

For the quadratic function represented by \(f(x)=x^{2}\), the domain will include all real numbers as the horizontal extent of the graph is the complete real number line.

Domain of Cubic Functions

For the cubic function represented by \(f(x)=x^{3}\), the domain will involve all real numbers as the horizontal length of the graph is the entire real number line.

Domain of Reciprocal Functions

For the reciprocal function represented by \(f(x)=\frac{1}{x}\), we cannot divide the function by zero, so we need to exclude zero from the domain.

The domain for such a function is given by: \((-\infty, 0)\cup (0, \infty)\).

Check out this article on Locus.

Classification of Types of Functions

In the functions and types of function, we were introduced to the notions of domain and range.

In mathematics, the domain of a function shows for which values of x the function is correct. This indicates that any value inside that domain will operate in the function, while any value that comes outside of the domain will not operate in the function.

In order to understand the meaning of domain in math, we must have an idea regarding the types of functions for easy understanding and learning. The types of functions have been classified into different categories, and are shown in the below table.

Based on Elements	One-One Function Many-One Function Onto Function One-One and Onto Function Into Function Constant Function
Based on the Equation	Identity Function Linear Function Quadratic Function Cubic Function Polynomial Functions
Based on the Range	Modulus Function Rational Function Signum Function Even and Odd Functions Periodic Functions Greatest Integer Function Inverse Function Smallest Integer Functions Composite Functions
Based on the Domain	Algebraic Functions Trigonometric Functions Logarithmic Functions Exponential Functions

Learn the various concepts of the Binomial Theorem here.

Domain of a Function: Key Takeaways

We can imagine the domain as a holding space that contains raw substances for a function machine and the range as another holding space for the machine's outcomes.

• Oftentimes while finding the domain of functions involves remembering three distinct forms.
• First, if the given function has no denominator or an even root, examine whether the domain could include all real numbers.
• Second, if there exists a denominator in the function's equation, eliminate the values of the domain that make the denominator to be zero.
• Third, if there are all even roots in the function, consider eliminating values that would cause the radicand to be negative.

Check out this article on Linear Inequality.

A function is a relationship in which every element of the input is linked with exactly one component of the output. A function relates the inputs to outputs. Below are some important key takeaways regarding the topic.

• Output: The output is the outcome or answer of a function.
• All the outputs all together are termed as the range.
• Relation: A relation is an association between numbers/symbols/characters in one set and numbers in another set.
•  A function drives elements from a set that is the domain and links them to elements in a set that is the codomain.
• Dependent variable: The dependent variable in a function is the one whose value depends on one or more independent variables of the given function.
• Independent variable: An independent variable in a function is one whose value does not depend on any other variable in the function.
• Graph: A diagram illustrating data; more precisely one explaining the relationship between two or more quantities, dimensions, or characters.
• A given relation is supposed to be a function if each of the elements of set A has one and only one image in set B.

If you are reading Domain of a Function, you can also read about the Matrices here.

Some functions for example the linear functions have domains that cover all potential values of x.

• Others such as equations where x arrives within the denominator eliminate certain values of x to avoid dividing by zero.
• Square root functions possess more limited domains than some other functions as the value inside the square root must be a positive number for the result to be a real one.

The conventions of interval notation that is followed while writing domain of a function are as follows:

• The smallest term from the interval is drafted first.
• The term larger than the smallest one in the interval is addressed second, followed by a comma and the process goes on for the rest of the numbers.
• Parentheses ( ) are applied to signify that an endpoint is not covered, termed exclusive.
• Brackets, [ ], are applied to show that an endpoint is involved, termed inclusive.

Also, read about Permutations and Combinations here.

The domain of a function can be arranged by placing the input values of a set of ordered pairs.

• The domain of a function can also be calculated by recognising the input values of a function written in an equation format.
• Interval values expressed on a number line can be drawn using inequality notation, set-builder notation, and interval notation.
• For many functions, the domain and range can be calculated from their respective graphs.
• A knowledge of toolkit functions can be practised to obtain the domain and range of relevant functions.

We hope that the above article on Domain of a Function is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

Domain of a Function FAQs

Q.1 What is the domain meaning in maths?

Ans.1 The domain in math can be taken as a set of the values that go inside a function; furthermore, the range implies all the values that come out.

Q.2 How to find the domain and range of a function?

Ans.2 The domain of a function can be arranged by placing the input values of a set of ordered pairs. The domain of a function can also be calculated by recognising the input values of a function written in an equation format.

Q.3 What do you mean by function?

Ans.3 A function relates to a specific relationship that outlines each element of one set with only one element relating to another set.

Q.4 What is a relation?

Ans.4 In mathematics, a relation describes the relationship between sets of values of ordered pairs. The set of components in the first set are termed as a domain that is related to the set of the component in another set, which is designated as the range.

Q.5 What are the domains and ranges of a function?

Ans.5 If f: P → Q is a function, then the set P is named as the domain of the function f. Similarly If f: P → Q is a function, then the range of f consists of those components of Q which are connected with at least one element of P.

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