Q.If $R_1 = \{(x, y) \mid y = 2x + 7,\ \text{where } x \in \mathbf{R} \text{ and } -5 \le x \le 5\}$ is a relation. Then find the domain and Range of $R_1$.

The relation $R_1$ is defined by the linear equation $y = 2x + 7$ with a restricted domain $-5 \le x \le 5$.

Domain: The domain is explicitly given as all real numbers $x$ satisfying $-5 \le x \le 5$. In interval notation, this is $[-5, 5]$.

Range: Since $y = 2x + 7$ is a strictly increasing linear function (positive slope), the minimum and maximum values of $y$ occur at the endpoints of the domain:

• At $x = -5$: $y = 2(-5) + 7 = -3$
• At $x = 5$: $y = 2(5) + 7 = 17$ …

Understanding Relations and Their Components

A relation $R_1$ is simply a collection of ordered pairs $(x, y)$ that satisfy a given condition. Think of it as a rule that pairs inputs with outputs. The domain is the set of all possible first coordinates (the $x$-values we're allowed to use), while the range is the set of all resulting second coordinates (the $y$-values we actually get).

In this problem, the relation is defined by a linear equation $y = 2x + 7$, but crucially, $x$ is restricted to the interval $[-5, 5]$. This restriction directly determines both the domain and the range.

Finding the Domain

The domain is straightforward here because it's explicitly given in the problem statement.

1. Identify the constraint on $x$.

We're told that $x \in \mathbf{R}$ and $-5 \le x \le 5$. This means $x$ can be any real number between $-5$ and $5$, inclusive.

2. Write the domain.

The domain of $R_1$ is simply the interval of allowed $x$-values:

Finding the Range

The range requires us to determine what $y$-values are produced when $x$ varies over the domain. Since $y = 2x + 7$ is a linear function with positive slope, it's strictly increasing—as $x$ increases, so does $y$.

3. Find the minimum value of $y$.

The smallest $y$ occurs at the smallest $x$. Substitute $x = -5$:

4. Find the maximum value of $y$. …