Question

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

Answer 1

Let $X=\{1,2,3,4\},Y=\{1,3,5,7\}$
BY definition of g we observe that every element in X has a unique image in Y and hence g is a function
$g\left(x\right)=ax+\beta$ when $x=1,g\left(x\right)=1$
and when $x=2,g\left(x\right)=3$
$\therefore 1=\alpha +\beta$ and $3=2\alpha +\beta .$ These give $\alpha =2,\beta =-1.$
$\therefore g\left(x\right)=2x-1$
for $x=3,g\left(x\right)=5$ and for $x=4,g\left(x\right)=7$ which are true.