Question

In the following, determine the value of $k$ so that the given function is continuous:

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

Answer 1

Given,

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

LHL at $x=3$,

$\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(3-h)=\underset{h\to 0}{lim}2=2$

RHL at $x=3$,

$\underset{x\to 3^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(3+h)=\underset{h\to 0}{lim}(a(3+h)+b)=3a+b$

LHL at $x=5$,

$\underset{x\to 5^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(5-h)=\underset{h\to 0}{lim}(a(5-h)+b)=5a+b$

RHL at $x=5$,

$\underset{x\to 5^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(5+h)=\underset{h\to 0}{lim}9=9$

Since $f\left(x\right)$ is continuous at $x=3$, therefore,

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

$\Rightarrow 3a+b=2.....\left(1\right)$

Also $f\left(x\right)$ is continuous at $x=5$, therefore,

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

$\Rightarrow 5a+b=9.....\left(2\right)$

Solving equation $\left(1\right)$ and $\left(2\right)$, we get

$a=\frac{7}{2}$ and $b=-\frac{17}{2}$