Question

If we have a relation as $R=\left\{ \left( x,y \right):y=\left| x-2 \right|;x\in z,\left| x \right|\le 3 \right\}$ find domain and range of the relation?

Answer 1

Hint: The above given question is a relation. A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x, y) is in relation.

Complete step by step solution:
The above given relation is:
$\Rightarrow R=\left\{ \left( x,y \right):y=\left| x-2 \right|;x\in z,\left| x \right|\le 3 \right\}$
Here we have to find the domain and range set of the given relation.
The possible value of x which satisfies the given relation is called domain of the relation.
The possible value of y which satisfies the given relation is called range of the relation.
If we look carefully in the question we see that $\left| x \right|\le 3$ , we know whenever we use this type expression we can write it as: $-3\le x\le 3$. It is the property of the mode function.
From this above expression we can concluded that the domain of the given relation is lies between $-3$ to $3$ or we can write by using closed interval as:
$\Rightarrow $ Domain = $\left\{ -3,-2,-1,0,1,2,3 \right\}$ Here the closed interval means $-3$ and $3$ are also included in the domain of the given relation.
Now to find the range of the given relation, we will use $\left\{ -3,-2,-1,0,1,2,3 \right\}$.
The given relation is:
$\Rightarrow y=\left| x-2 \right|$
Since the given x is an integer, we will put the values of the domain one by one to find the range of the given relation.
Now put $-3$ in the given relation we get
$\Rightarrow y=\left| -3-2 \right|=5$
Similarly if we put $-2,-1,0,1,2,3$ we get
$\begin{align}
  & \Rightarrow y=\left| -2-2 \right|=4 \\
 & \Rightarrow y=\left| -1-2 \right|=3 \\
 & \Rightarrow y=\left| 0-2 \right|=2,y=\left| 1-2 \right|=1,y=\left| 2-2 \right|=0,y=\left| 3-2 \right|=1 \\
\end{align}$
So the value of range is $\left\{ 0,1,2,3,4,5 \right\}$ So here we the above both functions are looking like a given function $y=\left| x-2 \right|$ and we know the value of y gives the range of the function. so we get that our given function lies between $5$ and $0$
So the range of the given relation is $\left\{ 0,1,2,3,4,5 \right\}$
Hence, the domain is $\left\{ -3,-2,-1,0,1,2,3 \right\}$ and range is $\left\{ 0,1,2,3,4,5 \right\}$.

Note: If we talk about the domain and range so in most of the question domain is given in the question we have to see the question very carefully. In the above question the domain is given in the form of $\left| x \right|\le 3$. To solve this function we should know everything about the mode function. And after the domain is determined, then we can easily find the range of any relation. Only we have to take care of where the values are coming from.