Question

Find the relationship between $a$ and $b$, so that the function $f\left(x\right)$ defined by $f\left(x\right)=$ $\{\begin{array}{c}ax+2,ifx\le 4\\ bx+4,ifx>4\end{array}$
is continuous at $x=3$

Answer 1

The given function is $f\left(x\right)=\{\begin{array}{c}ax+1,\text{if}x\le 3\\ bx+3,\text{If}x>3\end{array}$
If $f$ is continuous at $x=3$, then
$\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}f\left(x\right)=f\left(3\right)$ ....(1)
Now,
$\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{-}}{lim}(ax+1)=3a+1$
$\underset{x\to 3^{+}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}(bx+3)=3b+3$
and $f\left(3\right)=3a+1$
Therefore from (1), we obtain
$3a+1=3b+3=3a+1$
$\Rightarrow 3a+1=3b+3$
$\Rightarrow 3a=3b+2\Rightarrow a-b=\frac{2}{3}$
Therefore the required relationship is given by, $a-b=\frac{2}{3}$