Domain and Range of a Function
Consider the set A = {1, 2, 3, 4}. Let us define a function \(f\left( x \right) ={x^2}\) with the input set as the set A. Let us name the output set as set B. We thus have the following scenario:
The set A consists of all the input values, while the set B consists of all the output values. We now define the following two terms:

Domain of a function – this is the set of input values for the function. In the example above, the domain of \(f\left( x \right)\) is set A.

Range of a function – this is the set of output values generated by the function (based on the input values from the domain set). In the example above, the range of \(f\left( x \right)\) is set B.
Let’s take another example. Let X be the set {\(  1\) , 0, 1, 2}, while \(g\left( x \right)\) be a function defined as \(g\left( x \right) = {x^3}\). If we apply the function g on set X, we have the following picture:
The set X is the domain of \(g\left( x \right)\) in this case, whereas the set Y = {\( 1\), 0, 1, 8} is the range of the function corresponding to this domain.
When a function f has a domain as a set X, we state this fact as follows: f is defined on X.
Example 1: Let f be a function defined on \(\mathbb{Z}\) (the set of all integers), such that \(f\left( x\right) = {x^2}\). Find the domain and the range of f.
Solution: The domain of f has already been stated in the question: the set of all integers, \(\mathbb{Z}\) . Now, any integer when squared will generated a positive perfect square. Thus, the set of output values will be these.
\[{0, 1, 4, 9, 16, …}\]
We can thus say that the range is the set of all positive perfect squares. We can write this as follows:
\[R = {{n^2},\, n \;in\; \mathbb{Z} }\]
Note that since the domain is discrete, the range is also discrete.
Example 2: The plot of a function f is shown below:
Find the domain and range of the function.
Solution: We observe that the graph corresponds to a continuous set of input values, from \( 2\) to 3. Thus, the domain of the function is \(\left[ {  2,3} \right]\).Also, the variation in the function output is in the continuous interval from \( 1\) to 4. Thus, the range of the function is \(\left[ {  1,4} \right]\).
Example 3: Let f be a function defined on \(\left[ { 1,3} \right]\) such that \(f\left( x\right) = 2x  1\). Plot the graph of f and determine its domain and range.
Solution: The graph of f will be linear, as shown below:
The domain is clearly \(\left[ {  1,3} \right]\). Also, we note that the function takes all values in the continuous interval from \( 3\) to 5. Thus, the range of the function is \(\left[ {  3,5} \right]\).
Example 4: f is a function defined on \(\left[ { 2,1} \right]\) such that \(f\left( x\right) = \frac{1}{2}{x^2}\). Plot the graph of f, and find its domain and range.
Solution: First, we determine a few markers to aid us in our plotting process:
x 
\(  2\) 
\(  1\) 
0 
\(\frac{1}{2}\) 
1 
\(f\left( x \right)\) 
2 
\(\frac{1}{2}\) 
0 
\(\frac{1}{8}\) 
\(\frac{1}{2}\) 
Point 
\(\left( {  2,2} \right)\) 
\(\left( {  1,\frac{1}{2}} \right)\) 
\(\left( {0,0} \right)\) 
\(\left( {\frac{1}{2},\frac{1}{8}} \right)\) 
\(\left( {1,\frac{1}{2}} \right)\) 
Using these markers, the plot of f has been drawn below:
The domain of f is clearly \(\left[ {  2,1} \right]\). From the plot, it is clear that the range is \(\left[ {0,2}\right]\).
Example 5: What will be the range of the function \(f\left(x \right) = 1 + {x^2}\) if the domain is the set of all real numbers?
Solution: If x varies over all real numbers, then \({x^2}\) takes all values in the set \(\left[ {0,\infty } \right)\),because \({x^2} \ge 0\). Thus, \(1 + {x^2}\) takes all values in the set \(\left[ {1,\infty } \right)\). This means that the range of f is \(\left[ {1,\infty }\right)\). This is clear from the following figure, which shows the graph of \(f\left( x \right)\). Note the variation in output values – from a minimum of 1 towards infinity:
Example 6: The function \(f\left( x \right) = 2 + {x^3}\)is defined on a set X, and its range is Y = {\( 6\), 1, 2}. What is the domain of the function?
Solution: If \(f\left( x \right) =  6\), then \(2 + {x^3} =  6\), which means that \(x =  2\). Similarly, when \(f\left( x \right) = 1\), then \(x =  1\), and when \(f\left( x \right) = 2\), then \(x = 0\). We have the following map:
Thus, the domain of the function is
\[X = {\,(  2), ( 1), 0}\]