Question

Find the relationship between a and b so that the function f is defined by
f(x)=\left\\{\begin{matrix} ax+1 &if\,x\leq3 \\\ bx+3&if\,x>3 \end{matrix}\right.
is continuous at x=3.

Answer 1

f(x)=\left\\{\begin{matrix} ax+1 &if\,x\leq3 \\\ bx+3&if\,x>3 \end{matrix}\right.

If f is continuous at x=3,then
$\lim_{x\rightarrow -3}$ f(x)=$\lim_{x\rightarrow 3+}$f(x)=f(3) ...(1)
Also,
$\lim_{x\rightarrow 3^-}$ f(x)=$\lim_{x\rightarrow 3^-}$(ax+1)=3a+1
$\lim_{x\rightarrow 3^+}$ f(x)=$\lim_{x\rightarrow 3^+}$(bx+3)=3b+3
f(3)=3a+1
Therefore, from (1), we obtain
3a+1=3b+3=3a+1
⇒3a+1=3b+3
⇒3a=3b+2
⇒a=b+$\frac{2}{3}$
Therefore, the required relationship is given by,a=b+$\frac{2}{3}$