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+\(\frac{2}{3}\)