Q.If $R_3 = \{(x,\ |x|) \mid x \text{ is a real number}\}$ is a relation. Then find domain and range of $R_3$.

The key idea is to apply the definitions of domain and range to the given relation.

1. The domain of a relation is the set of all first components of its ordered pairs.
2. For the relation $R_3 = \{(x,\ |x|) \mid x \text{ is a real number}\}$, the first component of each ordered pair is $x$.
3. The condition given for $x$ is that $x$ is a real number, which means $x \in \mathbb{R}$. Therefore, the domain is $\mathbb{R}$.
4. The range of a relation is the set of all second components of its ordered pairs. …

When we talk about a relation, we're essentially describing a set of ordered pairs $(x, y)$. The first element, $x$, comes from a set called the domain, and the second element, $y$, comes from a set called the codomain. The actual values that $y$ takes on for the given $x$ values form the range.

Think of it like a machine: you put an input ($x$) into the machine, and it gives you an output ($y$).

• The domain is the collection of all valid inputs you can feed into the machine.
• The range is the collection of all possible outputs the machine can produce.

In this problem, our "machine" is the absolute value function, $y = |x|$. We need to figure out what real numbers we can put into this function and what real numbers can come out.

Let's break down the relation $R_3$:

1. Understand the Relation $R_3$

   The relation $R_3$ is defined as the set of all ordered pairs $(x, |x|)$ where $x$ is a real number. This means for any real number $x$ you choose, the corresponding $y$-value in the pair is its absolute value, $|x|$.

   So, we are looking at the function $f(x) = |x|$.

2. Determine the Domain of $R_3$

   The domain consists of all possible values for $x$. The definition states that $x$ is a real number. We need to check if there are any real numbers for which the expression $|x|$ is undefined.

   The absolute value function, $f(x) = |x|$, is defined for every real number $x$. You can take the absolute value of any positive number, any negative number, or zero. There are no restrictions like division by zero or taking the square root of a negative number.

   Therefore, the domain of $R_3$ is the set of all real numbers.

   In set-builder notation: Domain$(R_3) = \{x \mid x \in \mathbb{R}\}$

   In interval notation: Domain$(R_3) = (-\infty, \infty)$

3. Determine the Range of $R_3$

   The range consists of all possible values for $y$, which in this case are the values of $|x|$. Let's recall the definition of the absolute value:

Let's examine the possible outputs:
*   If $x$ is a positive number (e.g., $x=5$), then $|x|=x=5$. The output is positive. …