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

$\alpha = 2, \beta = -1$

The correct answer is Option (3) → $\alpha = 2, \beta = -1$ ##

Given that, $g = \{(1, 1), (2, 3), (3, 5), (4, 7)\}$.

Here, each element of domain has unique image. So, $g$ is a function.

Now given that,

$g(x) = \alpha x + \beta$

$g(1) = \alpha + \beta$

$\alpha + \beta = 1$ --- (i)

$g(2) = 2\alpha + \beta$

$2\alpha + \beta = 3$ --- (ii)

From Eqs. (i) and (ii), we get

$2(1 - \beta) + \beta = 3$

$\Rightarrow 2 - 2\beta + \beta = 3$

$\Rightarrow 2 - \beta = 3$

$\beta = -1$

If $\beta = -1$, then $\alpha = 2$

$\alpha = 2, \beta = -1$