Q.Is $g = \{(1, 1), (2, 3), (3, 5), (4, 7)\}$ a function? Justify. If this is described by the relation, $g(x) = \alpha x + \beta$, then what values should be assigned to $\alpha$ and $\beta$?

Concept: A relation is a function if each input maps to exactly one output; linear functions have the form $g(x) = \alpha x + \beta$.

Check if $g$ is a function:

Each first coordinate (1, 2, 3, 4) appears exactly once, so every input has a unique output. Yes, $g$ is a function.

Find $\alpha$ and $\beta$:

Using two points from the relation:

• From $(1, 1)$: $1 = \alpha(1) + \beta \implies \alpha + \beta = 1$
• From $(2, 3)$: $3 = \alpha(2) + \beta \implies 2\alpha + \beta = 3$ …

Is $g$ a function?

A relation is a function when every element in the domain (the set of first coordinates) is paired with exactly one element in the codomain (the set of second coordinates). The key word is "exactly one" — no input can have two different outputs, though two inputs may share the same output.

Looking at $g = \{(1, 1), (2, 3), (3, 5), (4, 7)\}$:

• Input $1$ maps to $1$ only
• Input $2$ maps to $3$ only
• Input $3$ maps to $5$ only
• Input $4$ maps to $7$ only

Each input appears exactly once and has a unique output. Yes, $g$ is a function.

Finding $\alpha$ and $\beta$

We're told that $g(x) = \alpha x + \beta$ describes this relation. This means the ordered pairs must satisfy a linear equation. We need to find the slope $\alpha$ and intercept $\beta$.

1. Use two points to set up equations

Take $(1, 1)$ and $(2, 3)$:

From $(1, 1)$:

From $(2, 3)$:

2. Solve the system

Subtract equation (i) from equation (ii):

$\alpha = 2$ …