Chapter 1.3, Problem 16E

To determine

To find: The new value that should be assigned to $f\left(1\right)$ to make the new extended function continuous at $x=1$ .

Answer to Problem 16E

The new value of $f\left(1\right)=2$ should be assigned.

Explanation of Solution

Given:

  $f\left(x\right)=\left\{\begin{array}{ll}{x}^2-1,\hfill & -1\le x<0\hfill \\ 2x,\hfill & 0<x<1\hfill \\ 1,\hfill & x=1\hfill \\ -2x+4,\hfill & 1<x<2\hfill \\ 0,\hfill & 2<x<3\hfill \end{array}\right.$

  

Find the new value that should be assigned to $f\left(1\right)$ to make the new extended function continuous at $x=1$ .

Calculation:

Find the new value that should be assigned to $f\left(1\right)$ :

  $\underset{x\to 1}{\lim }f\left(x\right)$

Left hand limit:

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

Right hand limit:

  $\begin{array}{c}\underset{x\to 1^{+}}{\lim }f\left(x\right)=\underset{x\to 1}{\lim }\left(-2x+4\right)\\ =2\end{array}$

Given that, $f\left(1\right)=1$

So, to make the function continuous at $x=1$ , the value of $f\left(1\right)=2$ should be

assigned.

Thus, the value of $f\left(1\right)=2$ should be assigned.