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. That is the collection of all the possible outputs of a function is understood as the range of the function.

We can also say that after substituting the domain, the complete set of all values achievable as outcomes of the dependent variable is said to be the range. A function in relations and functions is imagined as a rule, which provides a unique output for all input. Mapping or transformation is applied to express a function in mathematics .

With this article, we will learn what is the domain and range of a function, How to find them for exponential functions, trigonometric functions, absolute value function, square root function and so on with solved examples and more.

Definition Of Domain and Range

For functions, we input varying numbers and we receive new numbers as the outcome of the operation performed. The domain of a function signifies the inputs to the function and the range are the possible outputs for the given inputs.

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}.

Domain and Range 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 and range of a function either algebraically or by the graphical approach. To obtain the domain and range 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 and range.

Domain of a Function

A function relates an input to output, that is function links each element of a set with specifically one element of another set. The domain in math can be taken as a set of the values that go inside 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.

Range of a Function

The range of a function in set theory is the set of all its outputs. If f: X → Y is a function, then the range of f consists of those components of Y which are connected with at least one element of X. Here we would say that X is the domain of the function and Y is the co-domain of the given function. The output of this function becomes the range.

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.

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)}.

Here, will have the domain of the elements that go into the function and the range of a function that comes out of 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…}.

How to find Domain and Range?

There are different types of functions in mathematics namely: Logarithmic functions , many – one, bijective, onto, inverse functions, polynomial function, trigonometric and inverse trigonometric functions, modulus function, signum function, composite functions, constant function, identity function, greatest integer function, and their derivatives e.g derivatives of logarithmic functions, derivatives of algebraic functions, etc. One thing that should be kept in mind while determining the domain and range of a function 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:

• 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 is 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}

Let’s learn the domain and range of some special functions considering different types of functions.

Domain and Range 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 and the Range = (0, ∞). Consider the graph for the function f: \(2^{x}\).

Domain and Range of Trigonometric Functions

We can say that the value of the below trigonometric functions swings between -1 and 1 and it is defined for all real numbers. Therefore the Domain of such functions is the set R and the Range for such functions is [-1, 1].

Domain and Range of an Absolute Value Function

For the absolute value function represented by f(x)=|x|, there exists no limitation on x values. Thus the domain = R, and range = [0, ∞). Therefore, the domain of the absolute value function(also known as modulus function) comprises the collection of all real numbers. Although, because the absolute value is determined as a distance from zero, the output can simply be greater than or equivalent to zero.

Domain and Range 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.

Accordingly, the range of a square root function is the collection of all non-negative real numbers. Thus the domain for such a function=[0,∞) and the range =[0,∞). If we include imaginary numbers then things can get more complex, however in most cases, we are only required to consider real numbers.

Similarly, we can find the domain and range of a polynomial function including linear, quadratic, cubic, identity function and so on.

Graphs of Domain and Range

Another approach to identifying the domain and range of a function is by applying graphs. For the given graph below, the domain points to the collection of probable input values. That is, the domain of a graph includes all the input values displayed on the x-axis. Likewise, the range is the collection of possible output values, which are displayed on the y-axis.

If the graph resumes beyond the part of the graph we can notice, then the domain and range may be greater than the observable values. Let us consider an example to understand the same:

• For the above graph, we can mark that the horizontal spread of the graph is from –3 to 1 which comprises the domain.
• The vertical spread of the graph is from 0 to –4, therefore the range is [−4,0).
• The above mentioned are the basic steps involved in finding the domain and range of a function on a graph.

Domain and Range 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 and range 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.

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

The conventions of interval notation that is followed while writing domain and range 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.
• The domain and range of a function can be arranged by placing the input values of a set of ordered pairs.

• They can also be calculated by recognising the input values of a function written in an equation format.
• For many functions, the domain and range can be calculated from their respective graphs.
• A knowledge of toolkit functions can be practiced to obtain the domain and range of relevant functions.

Domain and Range of a Function Solved Examples

With all the knowledge of the domain and range, method of calculation for various types of function; let us now practice some more examples to understand the same.

Solved Example 1: For the given function obtain the domain and range.

Solution:

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.

Solved Example 2: Find the range and domain for the graph\(f(x)=\sqrt{x+4}\). given below:

Solution: For the given graph function; the domain is x≥−4 as x cannot be smaller than −4.

Also, we notice that the curve is either on or beyond the horizontal axis irrespective of the value of x. Therefore the range is y ≥ 0.

Solved Example 3:For the given below function obtain the domain and range.

Solution: For the function \(g(x) =x^{3}\) 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,…}.

Solved Example 4:For the given relation obtain the domain and range.

{(2, –4), (4, -5), (3, –2), (6, 1), (5, 7)}

Solution: A relation is termed as a set of ordered pairs with x and y coordinates.

Hence:

The domain of the relation {(2, –4), (4, -5), (3, –2), (6, 1), (5, 7)} is {2, 3, 4, 5, 6}.

The range of the relation {(2, –4), (4, -5), (3, –2), (6, 1), (5, 7)} is {-5, –4 –2, 1, 7}.

We hope that the above article on Domain and Range 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.