Question

Find the values of a and b so that the function f given by \[f\left( x \right) = \begin{cases}1 , & \text{ if } x \leq 3 \\ ax + b , & \text{ if } 3 < x < 5 \\ 7 , & \text{ if }  x \geq 5\end{cases}\] is continuous at x = 3 and x = 5.

Answer 1

Given: 

\[f\left( x \right) = \begin{cases}1, \text{ if } x \leq 3 \\ ax + b,  \text{ if }  3 < x < 5 \\ 7, \text{ if } x \geq 5\end{cases}\]

We have
(LHL at x = 3) = 

\[\lim_{x \to 3^-} f\left( x \right) = \lim_{h \to 0} f\left( 3 - h \right)\]

\[= \lim_{h \to 0} \left( 1 \right) = 1\]

(RHL at x = 3) = 

\[\lim_{x \to 3^+} f\left( x \right) = \lim_{h \to 0} f\left( 3 + h \right)\]

\[= \lim_{h \to 0} a\left( 3 + h \right) + b = 3a + b\]

(LHL at x = 5) = 

\[\lim_{x \to 5^-} f\left( x \right) = \lim_{h \to 0} f\left( 5 - h \right)\]

\[= \lim_{h \to 0} \left( a\left( 5 - h \right) + b \right) = 5a + b\]

(RHL at x = 5) = 

\[\lim_{x \to 5^+} f\left( x \right) = \lim_{h \to 0} f\left( 5 + h \right)\]

\[= \lim_{h \to 0} 7 = 7\]

If f(x) is continuous at x = 3 and 5, then 

\[\lim_{x \to 3^-} f\left( x \right) = \lim_{x \to 3^+} f\left( x \right) \text{ and }  \lim_{x \to 5^-} f\left( x \right) = \lim_{x \to 5^+} f\left( x \right)\]

\[\Rightarrow 1 = 3a + b . . . \left( 1 \right) \text{ and }  5a + b = 7 . . . \left( 2 \right)\]

On solving eqs. (1) and (2), we get

\[a = 3 \text{ and }  b = - 8\]