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

Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
At $x=3$
LHL=$\lim\limits_{\large x\to 3^-}(ax+1)$
$\quad\quad=3a+1.$
$f(3)=3a+1$
RHL=$\lim\limits_{\large x\to 3^+}(bx+3)$
$\quad\quad=3b+3.$
$f$ is continuous if LHL=RHL=f(3)
Step 2:
$3a+1=3b+3$
$3a-3b=3-1$
$3a-3b=2$
$3(a-b)=2$
$a-b=\large\frac{2}{3}$
$a=\large\frac{2}{3}$$+b$
For any arbitrary value of $b$ we can find the value of $a$ corresponding the value of $b$.