Find the relationship between a and b so that the function f defined by
Question:

Find the relationship between a and b so that the function f defined by

$f(x)= \begin{cases}a x+1, & \text { if } x \leq 3 \\ b x+3, & \text { if } x>3\end{cases}$

is continuous at = 3.

Solution:

The given function $f$ is $f(x)=\left\{\begin{array}{l}a x+1, \text { if } x \leq 3 \\ b x+3, \text { if } x>3\end{array}\right.$

If f is continuous at x = 3, then

$\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{+}} f(x)=f(3)$     …..(1)

Also,

$\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{-}}(a x+1)=3 a+1$

$\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3^{+}}(b x+3)=3 b+3$

$f(3)=3 a+1$

Therefore, from (1), we obtain

$3 a+1=3 b+3=3 a+1$

$\Rightarrow 3 a+1=3 b+3$

$\Rightarrow 3 a=3 b+2$

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

Therefore, the required relationship is given by,$a=b+\frac{2}{3}$

 

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