Finding the Domain and Range of a Function: Domain, in mathematics, is referred to as a whole set of imaginable values. These values are independent variables. In other words, in a domain, we have all the possible x-values that will make the function work and will produce real y-values. The range, on the other hand, is set as the whole set of possible yielding values of the depending variable, which in this case, is y (generally). 

Finding the domain requires determining the values of the independent variables (which is usually x) that have been allowed to use. At the bottom of the fraction, 0 is usually debarred or we generally avoid negative values that are found under the square root sign. The range of a function is considered as an array of possible y-values. Continue reading to learn more about the domain and range of a function.

Definition of a Function

The relation \(f\) from set \(A\) to set \(B\) is a function if every element of set \(A\) has only one image in set \(B\). It is a subset of \(A \times B\). Here, the relation \(R\) is a function from the set \(A\) to \(B\).

What is the Domain, Codomain, and Range of a Function?

The set of elements in \(A\) that are plugged into the function \(f\) is called the domain.

The set \(B\) that is a collection of possible outcomes is called the codomain. The set of images of the elements in \(A\), which is a subset of \(B\) is called the range of the function \(f\)

Range \(\left\{ {y \in Y,y = f\left( x \right),\,x \in X} \right\}\)

For the function \(R:\)

Domain \( = ~\left\{ {1,~2,~3} \right\}\)

Codomain \( = ~\left\{ {5,~6,~7,~8} \right\}\)

Range \( = ~\left\{ {5,~6,~8} \right\}\)

How to Find the Domain of a Function?

The domain of a function is the values for which the function is defined.

For real-valued functions: first, you need to identify the values for which the function is not defined and then exclude them.

Example: A logarithmic function \(f(x)=\log x\) is defined only for positive values of \(x\). That is, the domain of the function is the set of positive real numbers. So, that is how it, i.e., domain and range of logarithmic functions, works.

How to Find the Range of a Function?

The range is the set of images of the elements in the domain.

To find the range of a function:

• Step 1: Write down the function in the form \(y=f(x)\)
• Step 2: Solve it for \(x\) to write it in the form, \(x=g(y)\)
• Step 3: The domain of the function \(g(y)\) is the range of \(f(x)\).

Example: For the function \(f(x)=\log x\), the image takes up the values from \(-\infty\) to \(+\infty\). That is, the range of the function is the set of all real numbers.

Find the Domain and Range from Graphs

We know that the domain of a function is the set of all input values. So, the domain on a graph is all the input values shown on the \(x\)-axis. To find the domain, we need to analyse what the graph looks like horizontally. Moving from left to right along the \(x\)-axis, identify the span of values for which the function is defined.

Similarly, the range of a function is the set of all output values. On a graph, this is identified as the values that are taken by the dependent variable \(y\). So, to find the range, look at the set of values that the graph spreads vertically. Those looking for the domain and range calculator should take help from the figures shown on this page.

Consider the graph of the function \(y=\sin x\).

Looking at the horizontal and vertical spread of the graph, the domain, and the range can be calculated as shown below.

The closed points on either end of the graph indicate that they are also part of the graph. Therefore, the domain is \( – \pi \le x \le \pi ,\) and the range is \(-1 \leq y \leq 1\).

Now, if you have open points instead, the function is not defined at that point.

Here, the domain is \( – \pi \le x < \pi ,\) and the range is \(-1 \leq y \leq 1\). Note that, here the value \(y=0\) is included in the range as it already has a pre-image at \(x = \, – \pi \) and \(x=0.\)

Sometimes the graph continues beyond the portion shown. In such cases, the domain and range could be greater than the visible values.

Generally, the arrows on either end show that the graph extends infinitely in both directions, and hence, the domain is the set of all real numbers. However, the range in this particular case remains the same as \(-1 \leq y \leq 1\)

Find the Domain and Range from Equations

Let \(y=f(x)\) be the function we need to find the domain and the range.

Step 1: Solve the equation to determine the values of the independent variable \(x\) and obtain the domain.

Step 2: To calculate the range, rewrite the equation \(y=f(x)\) with the independent variable \(x\) expressed in terms of \(y\). That is, in the form \(x=g(y)\). Now, the domain of the function \(g(y)\) is the range of the function \(f(x)\).

Find the Domain and Range of Special Functions

Rational Function: A rational function is defined for only the non-zero values of the denominator.

1. Equate the denominator to zero and solve for \(x\) to find the values to be excluded.
2. Once the values are excluded in the domain, the range is calculated by excluding the images of those values. 

Square Root Function: A square root function is defined for only the non-negative values of the expression under the radical symbol. 

1. Find the excluded values for \(x\).
2. The range is calculated by omitting the images of the excluded values in the domain.

Solved Examples

Q.1. A function \(f(x)=3 x\) is defined from set \(A\) to set \(B\) where \(A = ~\left\{{1,~2,~3,~4,~5} \right\}\) and \(B = ~\left\{{0,~1,~2,~3,~4,~5,~6,~7,~8,~9,~10,~11,~12,~13,~14,~15,~16}\right\}\). What are the domain and range of the function \(f\) ? Are range and codomain the same?
Ans:
Domain, \(A = ~\left\{{1,~2,~3,~4,~5} \right\}\)
Codomain, \(B = ~\left\{{0,~1,~2,~3,~4,~5,~6,~7,~8,~9,~10,~11,~12,~13,~14,~15,~16}\right\}\)
Range is the set of all \(f(x)\)’s for every \(x∈A\)

\(f(1)=3\)

\(f(2)=6\)

\(f(3)=9\)

\(f(4)=12\)

\(f(5)=15\)

\(\therefore\) Range, \(B = ~\left\{{3,~6,~9,~12,~15} \right\}\)
Hence, range \(\neq\) codomain.

Q.2. Find the domain of \(f(x)=\frac{x^{2}+2 x+1}{x^{2}+3 x+2}\).
Ans: Given: \(f(x)=\frac{x^{2}+2 x+1}{x^{2}+3 x+2}\)
\(=\frac{(x+1)^{2}}{(x+1)(x+2)}\)
\(=\frac{x+1}{x+2}\)
Since a rational function is defined only for non-zero values of its denominator, we have,
\(x+2 \neq 0\)
\(\Rightarrow x \neq-2\)
\(\therefore\) Domain \( = \left\{ {x \in R,x \ne – 2} \right\}\)

Q.3. What are the domain and range of the real-valued function \(g(x)=\sqrt{75-x^{2}}\) ?
Ans: A square root function is defined only for non-negative values under the square root symbol.
Given: \(75-x^{2} \geq 0\)
\(\Rightarrow 75 \geq x^{2}\) or \(x^{2} \leq 75\)
\(-\sqrt{75} \leq x \leq \sqrt{75}\)
\(\therefore\) Domain \(={x \in R,-\sqrt{75} \leq x \leq \sqrt{75}}\) or \([-\sqrt{75}, \sqrt{75}]\)
Let \(y=\sqrt{75-x^{2}}\)
\(y^{2}=75-x^{2}\)
\(x^{2}=75-y^{2}\)
Since \(x \in[-\sqrt{75}, \sqrt{75}]\), the value of \(y\) varies from \(0\) to \(\sqrt{75}\)
\(\therefore\) Range \(={y \in R, 0 \leq y \leq \sqrt{75}}\) or \([0, \sqrt{75}]\)

Q.4. Find the domain for which the functions \(f(x)=2 x^{2}+3 x+1\) and \(g(x)=x^{2}-5 x-14\) are equal.
Ans:
Given: \(f(x)=g(x)\)
\(\therefore 2 x^{2}+3 x+1=x^{2}-5 x-14\)
\(2 x^{2}+3 x+1-x^{2}+5 x+14=0\)
\(x^{2}+8 x+15=0\)
\((x+3)(x+5)=0\)
\(x=-3\) or \(x=-5\)
Therefore, the domain for which the functions \(f(x)\) and \(g(x)\) are equal is \(\left\{ { – 3,\, – 5} \right\}\)

Q.5. Identify the domain and range of the function represented by the graph.

Ans: The open dot on the left extreme shows that the plotted point is not included. That is, the function is not defined for the point \(x=-1\).
From the graph, the function is defined for all the values from \(-1\) to \(3\), including \(3\) and excluding \(-1\).
So, the domain of the function is \(-1<x \leq 3\)
The range varies from \(0\) to \(4\), including both points.
That is, the range is \(0 \leq y \leq 4\).

Domain: \({x \in R,-1<x \leq 3}\) or \((-1,3]\)
Range: \({x \in R, 0 \leq x \leq 4}\) or \([0,4]\)

Summary

The article defines a function, its domain, range, and codomain. It goes on to explain each in detail with examples. Further, it explains the methods to find the domain and range of a function when the function rule or an equation, or the graph of a function is given.

The article also discusses the key points in finding the domain and range of some special functions such as rational and square root functions. It concludes with a few solved examples to have emphasised the idea of the concepts and the calculations involved. Those searching for the domain and range table should read the article on our website.

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FAQs on Finding the Domain and Range of a Function

We have added the domain and range examples below so that aspirants can understand different aspects of the same and get resolutions to their queries.

Q.1. What is the difference between domain and codomain?
Ans: When a function \(f\) is defined from set \(X\) to set \(Y\):
1. The set of elements in \(X\) that are plugged into the function \(f\) is called the domain.
2. The set \(B\) that is a collection of possible outcomes is called the codomain. 

Q.2. How to find the domain and range of a function algebraically?
Ans: To find the domain of a function, find the values for which the function is defined. For example, a rational function is defined for only the non-zero values of the denominator. So, equate the denominator to zero and solve for \(x\) to find the values to be excluded.
Now, to find the range of the function, write down the function in the form \(y=f(x)\) and solve it for \(x\) to write it in the for \(x=g(y)\). Now the domain of the function \(g(y)\) is the range of the function \(f(x)\).

Q.3. How do you find the domain and range of a function without graphing? 
Ans: Let \(f(x)\) be the function to find the domain and the range.
Step 1: Rewrite the equation representing the function in the form \(y=f(x)\).
Step 2: Solve the equation to determine the values of the independent variable \(x\) to obtain the domain.
Step 3: Rewrite the equation \(y = f(x)\) with the independent variable x expressed in terms of \(y\). That is, in the form \(x=g(y)\).
Step 4: The domain of the function \(g(y)\) is the range of \(f(x)\).

Q.4. What is the domain and range in a graph? 
Ans: The domain of a function is the set of input values. So, the domain in a graph is the input values shown on the \(x\)-axis. The range of a function is the set of the output values. On a graph, this can be identified as the values taken by the dependent variable \(y\). Therefore, on a graph, the domain and range can be found by identifying the range of \(x\) and \(y\)-value variations.

Q.5. How do you find the domain and range of a quadratic equation?
Ans: Consider the parent quadratic function \(f(x)=x^{2}\).

Observe the graph of the given quadratic equation. Moving from left to right along the \(x\)-axis, there are no holes in the graph specifying the points where the function is not defined. So, the domain is the set of all real numbers.

The range of the function is the values that the graph spreads vertically. The \(y\)-values are all greater than or equal to zero. Thus, the range is a set of all real numbers greater than or equal to zero.
Since all other quadratic functions are transformations of the parent function, their domain and range can be calculated as transformations of this function.

We hope this detailed article on finding the domain and range of a function helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!