Chapter 2.3, Problem 48E

To determine

To find: The value of $a$ such that the function $f\left(x\right)=\{\begin{array}{l}2x+3,x\le 2\\ ax+1,x>2\end{array}$ is continuous.

Answer to Problem 48E

The value of $a$ such that the function $f\left(x\right)=\{\begin{array}{l}2x+3,x\le 2\\ ax+1,x>2\end{array}$ is continuous is equal to $3$.

Explanation of Solution

Given information: The function $f\left(x\right)=\{\begin{array}{l}2x+3,x\le 2\\ ax+1,x>2\end{array}$ is continuous.

Calculation:

The function $f\left(x\right)=\{\begin{array}{l}2x+3,x\le 2\\ ax+1,x>2\end{array}$ is continuous. So, the function is continuous at all the points of $x$ . The function is also continuous at $x=2$ . So, $\underset{x\to 2}{\lim }f\left(x\right)=f\left(2\right)$.

The function $f\left(x\right)=2x+3$ corresponds to all the values of $x\le 2$.

Calculate the left hand limit of function at $x=2$.

  $\begin{array}{c}\underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }\left(2x+3\right)\\ =2\times 2+3\\ =4+3\\ =7\end{array}$

The function $f\left(x\right)=ax+1$ corresponds to all the values of $x>2$.

Calculate the right hand limit of function at $x=2$.

  $\begin{array}{c}\underset{x\to 2^{+}}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }\left(ax+1\right)\\ =a\underset{x\to 2^{+}}{\lim }\left(x\right)+\underset{x\to 2^{+}}{\lim }\left(1\right)\\ =a\times 2+1\\ =2a+1\end{array}$

It is given that the function is continuous at $x=3$ . So, left hand limit will be equal to right hand limit of the function as:

  $\begin{array}{c}\underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }f\left(x\right)\\ 7=2a+1\\ 2a=6\\ a=3\end{array}$

Therefore, the value of $a$ is 3 such that the function $f\left(x\right)=\{\begin{array}{l}2x+3,x\le 2\\ ax+1,x>2\end{array}$ is continuous.