Question

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

Answer 1

Given: 

 \[f\left( x \right) = \begin{cases}2 , & \text{ if }  x \leq 3 \\ ax + b, & \text{ if }  3 < x < 5 \\ 9 , & \text{ if }  x \geq 5\end{cases}\]

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

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

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