Question

Find the relationship between $a$ and $b$ so that the function $f$ defined by
$f\left(x\right)=\{\begin{array}{cc}ax+1,& ifx\le 3\\ bx+3,& ifx>3\end{array}$
is continuous at $x=3$.

Answer 1

Given:

$f\left(x\right)=\{\begin{array}{cc}ax+1,& ifx\le 3\\ bx+3,& ifx>3\end{array}$
We have $f\left(3\right)=3a+1$

$\therefore \underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}f\left(x\right)=f\left(3\right)$

$LHL=\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{-}}{lim}(ax+1)=3a+1$

$RHL=\underset{x\to 3^{+}}{lim}=\underset{x\to 3^{+}}{lim}(bx+3)=3b+3$

$\therefore 3a+1=3b+3$
$\Rightarrow 3a=3b+2$
$a=b+\frac{2}{3}$.