Question

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

Answer 1

Hint: By making use of the definition of function, first find out if it is a function or not. Then, find out the values of $\alpha $ and $\beta $

Complete Step-by-Step Solution:
Given $g = \{ (1,1),(2,3),(3,5),(4,7)\} $
Here, if we observe in the given set each element of the domain has a unique image so from this we can say the g is function.
Here the function g is described by a formula $g(x) = \alpha x + \beta $
Now let us substitute $x = 1$ in the formula $g(x) = \alpha x + \beta $
$\
  g(1) = \alpha (1) + \beta \\
  g(1) = \alpha + \beta \to 1 \\
\ $
We know that $g(1) = 1 $ from the given function g
So we write the equation as
$\alpha + \beta = 1 \to 1 $
Now again let us substitute $x = 2$ in the formula $g(x) = \alpha x + \beta $
$\
  g(2) = \alpha (2) + \beta \\
  g(2) = 2\alpha + \beta \\
\ $
And we know that $g(2) = 3 $ from the given function g
So we can write the equation as
$2\alpha + \beta = 3 $$ \to 2$
Now let us solve equation 1 and 2
We get $ \alpha = 2 $ and $ \beta = 1$
Now we can write the formula as $g(x) = 2x - 1$
$\therefore g(x) = 2x - 1$

Note: When solving these types of problems, first find the respective values which are needed and then find out the actual function (g(x) in this case).