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 $\beta$?
Answer & explanation
Correct answer: option 3
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 $\alpha = 2$
$\alpha = 2, \beta = -1$