Question

In the following, determine the value of constant involved in the definition so that the given function is continuou:  $$f\left(x\right)=\{\begin{array}{ll}2,& \text{ if }x\le 3\\ ax+b,& \text{ if }3<x<5\\ 9,& \text{ if }x\ge 5\end{array}$$

Answer 1

Given: 

 $$f\left(x\right)=\{\begin{array}{ll}2,& \text{ if }x\le 3\\ ax+b,& \text{ if }3<x<5\\ 9,& \text{ if }x\ge 5\end{array}$$

 if $$f\left(x\right)$$ is continuous at x = 3 and 5, then 

$$\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}f\left(x\right)\text{ and }\underset{x\to 5^{-}}{lim}f\left(x\right)=\underset{x\to 5^{+}}{lim}f\left(x\right)$$

$$\Rightarrow \underset{h\to 0}{lim}f(3-h)=\underset{h\to 0}{lim}f(3+h)\text{ and }\underset{h\to 0}{lim}f(5-h)=\underset{h\to 0}{lim}f(5+h)$$
$$\Rightarrow \underset{h\to 0}{lim}\left(2\right)=\underset{h\to 0}{lim}(a(3+h)+b)\text{ and }\underset{h\to 0}{lim}(a(5-h)+b)=\underset{h\to 0}{lim}\left(9\right)$$
$$\Rightarrow 2=3a+b\text{ and }5a+b=9$$
$$\Rightarrow 2=3a+b\text{ and }5a+b=9$$
$$\Rightarrow a=\frac{7}{2}\text{ and }b=\frac{-17}{2}$$