Domain and Range of a Function

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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:

Function's domain and range illustration

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:

Function's domain and range exam 

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:

Graph of a function

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:

Graph of linear function

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 has been drawn below:

Graph of function example 1

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:

Graph of function example 2

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:

Function's domain and range example 2

Thus, the domain of the function is

\[X = {\,( - 2), ( -1), 0}\]

Download practice questions along with solutions for FREE:
Functions
grade 10 | Answers Set 1
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Functions
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Download practice questions along with solutions for FREE:
Functions
grade 10 | Answers Set 1
Functions
grade 10 | Questions Set 2
Functions
grade 10 | Answers Set 2
Functions
grade 10 | Questions Set 1
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