Domain and Range are the two main factors of Function. The domain of a function is the set of input values of the Function, and range is the set of all function output values. A function is a relation that takes the domain’s values as input and gives the range as the output.

The primary condition of the Function is for every input, and there is exactly one output. This article will discuss the domain and range of functions, their formula, and solved examples.

Definition of Functions and Relations

Functions are one of the key concepts in mathematics which have various applications in the real world. Functions are special types of relations of any two sets. A relation describes the cartesian product of two sets.

Cartesian product of two sets \(A\) and \(B\), such that \(a \in A\) and \(b \in B\), is given by the collection of all order pairs \((a, b)\). Relation tells that every element of one set is mapped to one or more elements of the other set. We can say relation has for every input there are one or more outputs.

The function is the special relation, in which elements of one set is mapped to only one element of another set. A function is a relation in which there is only one output for every input value.

Consider a relation \(f\) from set \(A\) to set \(B\). And, a relation \(f\) is said to be a function of each element of set \(A\) is associated with only one element of the set \(B\).

Domain and Range of Function

The function is the relation taking the values of the domain as input and giving the values of range as output. All of the values that go into a function or relation are called the domain. All of the entities or entries which come out from a relation or a function are called the range.

The set of all values, taken as the input to the function, is called the domain. The values of the domain are independent values. The set of all values, which comes as the output, is known as the function’s range. The value of the range is dependent variables.

Example: The function \(f(x)=x^{2}\):

The values \(x=1,2,3,4, \ldots\) are the inputs and the values \(f(x)=1,4,9,16, \ldots\) are the output values.

Notation of Domain and Range of Function

The domain and range of the function are usually expressed in interval notation. Let us discuss the concepts of interval notations:

1. The smallest number should be written in the interval first
2. The largest number is written second in the interval, following comma
3. Parenthesis or \(()\) signifies that endpoints are not included; it is also known as exclusive.
4. Brackets or \([ ]\) are used to signify that endpoints are included; it is also known as inclusive.

The following table gives the different types of notations used along with the graphs for the given inequalities.

Finding the Domain of a Function

The domain of the function, which is an equation:

1. Identify the input values.
2. Identify any uncertainty on the input values. 
3. Exclude the uncertain values from the domain.
4. Write down the domain in the interval form.

The domain of the function, which is in fractional form, contains equation:

1. Identify the input values.
2. Identify any uncertainty on the input values. 
3. If there is a denominator in the function, make the denominator equal to zero and solve for the variable. 
4. Exclude the uncertain values from the domain.
5. Write the domain in interval form.

The domain of the function, which contains an even number of roots: 

1. Identify the input values.
2. Identify any uncertainty on the input values. 
3. If the given function contains an even root, make the radicand greater than or equal to 0, and then solve for the variable.
4. Exclude the uncertain values from the domain.
5. Write the domain in interval form.

Graph of Domain and Range of Functions

We know that all of the values that go into a function or relation are called the domain. All of the entities or entries which come out from a relation or a function are called the range.

We can find the domain and range of any function by using their graphs. The independent values or the values taken on the horizontal axis are called the function’s domain. The dependent values or the values taken on the vertical line are called the range of the function.

For the function: \(=f(x)\), the values of \(x\) are the domain of the function, and the values of \(y\) are the range of the function.

Domain and Range of Constant Function

For all values of the input, there is only one output, which is constant, and is known as a constant function. For the constant function: \(f(x)=C\), where \(C\) is any real number.

The input values of the constant function are any real numbers, and we can take there are infinite real numbers. So, the domain of the constant function is \((-\infty, \infty)\). The output of the given constant function is always constant \(‘C^{\prime}\). So, the range of the constant function is \(C\).

\({\text{Domain}}:( – \infty ,\infty );{\text{Range}}:{\text{C}}\)

Domain and Range of Identity Function

A function \(f(x)=x\) is known as an Identity function. For an identity function, the output values are equals to input values. All the real values are taken as input, and the same real values are coming out as output. So, the range and domain of identity function are all real values.

\({\text{Domain}}:( – \infty ,\infty );{\text{Range}}:( – \infty ,\infty )\)

Domain and Range of Absolute Value Function

The function \(f(x)=|x|\) is called absolute value function. For the absolute value function, we can always get positive values along with zero for both positive and negative inputs. So, all real values are taken as the input to the function and known as the domain of the function. The output values of the absolute function are zero and positive real values and are known as the range of function.

\({\text{Domain}}:( – \infty ,\infty );{\text{Range}}:[0,\infty )\)

Domain and Range of Quadratic Function

The function \(f(x)=x^{2}\), is known as a quadratic function. The graph of the quadratic function is a parabola. The output values of the quadratic equation are always positive. We can take any values, such as negative and positive real numbers, along with zero as the input to the quadratic function.

So, all the real values are the domain of the quadratic function, and the range of the quadratic function is all positive real values, including zero.

\({\text{Domain}}:( – \infty ,\infty );{\text{Range}}:[0,\infty )\)

Domain and Range of Cubic Function

The function, \(f(x)=x^{3}\), is known as cubic function. We know that, for a cubic function, we can take all real numbers as input to the function. The output of the cubic function is the set of all real numbers. For the negative values, there will be negative outputs, and for the positive values, we will get positive values as output.

So, the range and domain of the cubic function are set of all real values.

\({\text{Domain}}:( – \infty ,\infty );{\text{Range}}:( – \infty ,\infty )\)

Domain and Range of Exponential Function

The function, \(f(x)=a^{x}, a \geq 0\) is known as an exponential function.

Let us take an example: \(f(x)=2^{x}\). The graph of the function \(f(x)=2^{x}\) is given below:

\({\text{Domain}}:( – \infty ,\infty );{\text{Range}}:(0,\infty )\)

Here, the exponential function will take all the real values as input. The exponential function always results in only positive values. From the graph, we can observe that the graph comes closer to zero but never intersects at zero.

Domain and Range of Reciprocal Function

The function \(f(x)=\frac{1}{x}\) is known as reciprocal function. We know that the denominator of any function can not be equal to zero. The reciprocal function will take any real values other than zero. And similarly, the output values also any real values except zero.

So, the range and domain of the reciprocal function is a set of real numbers excluding zero.

\({\text{Domain}}:( – \infty ,0) \cup (0,\infty );{\text{Range}}:( – \infty ,0) \cup (0,\infty )\)

Domain and Range of Trigonometric Functions

The domain and range of trigonometric ratios such as sine, cosine, tangent, cotangent, secant and cosecant are given below:

Solved Examples

Q.1. Find the domain for the function \(f(x)=\frac{x+1}{3-x}\).
Ans:
Given function is \(f(x)=\frac{x+1}{3-x}\).
Solve the denominator \(3-x\) by equating the denominator equal to zero. \(3-x=0\)
\(\Longrightarrow x=3\)
Hence, we can exclude the above value from the domain.
Thus, the domain of the above function is a set of all values, excluding \(x=3\).
The domain of the function \(f(x)\) is \(R-{3}\).

Q.2. Find the domain and range of \(f(x)=\sin x\).
Ans:
Given function is \(f(x)=\sin x\).
The graph of the given function is given as follows:

From the above graph, we can say that the value of the sine function oscillates between \(1\) and \(-1\) for any value of the input. So, for any real values, the output of the sine function is \(1\) and \(-1\) only.
Domain of \(f(x)=\sin x\) is all real values \(R\) and range of \(f(x)=\sin x\) is \([-1,1]\).

Q.3. What is the range and domain of the function \(f(x)=\frac{1}{x^{2}}\) ?
Ans:
Given function is \(f(x)=\frac{1}{x^{2}}\).
The graph of the above function can be drawn as follows:

We know that denominator of the function can not be equal to zero. So, exclude the zero from the domain. So, the domain of the given function is a set of all real values excluding zero.
From the above graph, we can observe that the output of the function is only positive real values. The range of the given function is positive real values.

\({\text{Domain}}:( – \infty ,0) \cup (0,\infty );{\text{Range}}:(0,\infty )\)

Q.4. Find the range of the function \(f\left( x \right) = \{ \left( {1,~a} \right),~\left( {2,~b} \right),~\left( {3,~a} \right),~\left( {4,~b} \right)\).
Ans:
Given function is \(f\left( x \right) = \{ \left( {1,~a} \right),~\left( {2,~b} \right),~\left( {3,~a} \right),~\left( {4,~b} \right)\).
In the ordered pair \((x, y)\), the first element gives the domain of the function, and the second element gives the range of the function.
Thus, in the given function, the second elements of all ordered pairs are \(a, b\).
Hence, the range of the given function is \(\left\{ {a,~b}\right\}\).

Q.5. Identify the values of the domain for the given function:

Ans: We know that the function is the relation taking the values of the domain as input and giving the values of range as output.
From the given function, the input values are \(2,3,4\).
Hence, the domain of the given function is \(\left\{{2,~3,~4}\right\}\).

Summary

In this article, we studied the difference between relation and functions. We discussed what domain and range of function are. This article gives the idea of notations used in domain and range of function, and also it tells how to find the domain and range.

This article discussed the domain and range of various functions like constant function, identity function, absolute function, quadratic function, cubic function, reciprocal function, exponential function, and trigonometric function by using graphs.

FAQs

Q.1. How do you write the domain and range?
Ans: The domain and range are written by using the notations of interval.
1. Parenthesis or \(()\) is used to signify that endpoints are not included.
2. Brackets or \([ ]\) is used to signify that endpoints are included.

Q.2. What is the range on a graph?
Ans: The values are shown on the vertical line, or \(y\)-axis are known as the values of the range of the graph of any function.

Q.3. What is the difference between domain and range?
Ans: The domain is the set of input values to the function, and the range is the set of output values to the function.

Q.4. Explain Domain and Range of Functions with examples.
Ans: The set of all values, which are taken as the input to the function, are called the domain. The values of the domain are independent values. The set of all values, which comes as the output, is known as the range of the function. The value of the range is dependent variables.
Example: The function \(f(x)=x^{2}\):
The values \(x=1,2,3,4, \ldots\) are domain and the values \(f(x)=1,4,9,16, \ldots\) are the range of the function.

Q.5. What is the range of \(f(x)=\cos x\) ?
Ans: The range of the \(f(x)=\cos x\) is \([-1,1]\).

We hope this detailed article on domain and range of functions helped you. If you have any doubts or queries, feel to ask us in the comment section. Happy learning!