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

Given: g = {(1, 1), (2, 3), (3, 5), (4, 7)}, and it is described by relation g (x) = αx + β

To find: whether g is a function, and also to find the values of α and β

Explanation: the given relation is g = {(1, 1), (2, 3), (3, 5), (4, 7)}

Here for every element in the domain has the unique image.

And a relation is said to be function if every element of one set has one and only one image in other set.

So g is a function.

Now given the relation g = {(1, 1), (2, 3), (3, 5), (4, 7)} as

g (x) = αx + β

for ordered pair (1,1), g (x) = αx + β, becomes

g (1) = α(1) + β = 1

⇒ α + β = 1

⇒ α = 1-β………..(i)

Now consider other ordered pair (2, 3), g (x) = αx + β, becomes

g (2) = α(2) + β = 3

⇒ 2α + β = 3

Now substituting value of α from equation (i), we get

⇒ 2(2) + β = 3

⇒ β = 3-4 = -1

Now substituting the value of β in equation (i), we get

α = 1-β = 1-(-1) = 2

Hence the values 2 and -1 should be assigned to α and β to satisfy the given condition g (x) = αx + β, i.e., g (x) = 2x-1