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Is $g=\{(1,1),(2,3),(3,5),(4,7)$,$\} a function, justify. If this is$ described by the relation, $g(x)=\alpha x+\beta$, then what values should be assigned to $\alpha$ and $\beta$ ?
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We have, $g=\{(1,1),(2,3),(3,5),(4,7)\}$
Here, every element has unique image under $g$. So, $g$ is a function.
Now, $g(x)=\alpha x+\beta$
when $x=1$, then $g(1)=\alpha(1)+\beta$ $\Rightarrow 1=\alpha+\beta$
When $x=2$, then $g(2)=\alpha(2)+\beta \Rightarrow 3=2 \alpha+\beta$
On solving Eqs. (i) and (ii), we get $\alpha=2, \beta=-1$
Here, every element has unique image under $g$. So, $g$ is a function.
Now, $g(x)=\alpha x+\beta$
when $x=1$, then $g(1)=\alpha(1)+\beta$ $\Rightarrow 1=\alpha+\beta$
When $x=2$, then $g(2)=\alpha(2)+\beta \Rightarrow 3=2 \alpha+\beta$
On solving Eqs. (i) and (ii), we get $\alpha=2, \beta=-1$
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