Chapter 13, Problem 6CRT

a.

To determine

To graph: the function.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\left\{\begin{array}{l}3,\text{ if }x<0\\ 2\text{, if }x=0\\ 3-x,\text{ if 0}<x<2\\ x\text{ if }x\ge 2\end{array}\right\}$.

graph: to graph the function, use the point plotting method,

$x$	$f\left(x\right)$
$-1$	$3$
$0$	$2$
$1$	$2$

By the using of this table,

The graph can be obtained as:

  

Interpretation: form the above graph it can be observed that the function is not continuous.

b.

To determine

To find: the values of function.

Answer to Problem 6CRT

  $f\left(0\right)=2$

  $\underset{x\to 0}{\lim }f\left(x\right)=\text{does not exist}$

  $\underset{x\to 1}{\lim }f\left(x\right)=2$

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

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)=2$

Explanation of Solution

Given information:

  $f\left(0\right)$

  $\underset{x\to 0}{\lim }f\left(x\right)$

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

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

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)$

Calculation: from the graph in part (a) it can be observed that,

  $f\left(0\right)=2$

  $\underset{x\to 0}{\lim }f\left(x\right)=\text{does not exist}$

  $\underset{x\to 1}{\lim }f\left(x\right)=2$

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

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)=2$