Chapter 3.8, Problem 29PPS

To determine

To graph: the given function, to state the domain and range and then to describe the transformation of the given function related to $f(x)=\left|x\right|$ .

Explanation of Solution

Given information:

The function $g(x)=|x+2|+3$

Concept used:

Domain is the set of input values for which the function is defined.

Range is the set of output values for input values which are in domain.

Consider $g(x)=|x+2|+3=\{\begin{array}{l}-x+1;x<-2\\ x+5;x\ge -2\end{array}$

Make a table of values for $g(x)=|x+2|+3$

$x$	$-4$	$-2$	$0$	$10$	$15$
$g(x)$	$5$	$3$	$5$	$15$	$20$

The graph of $g(x)=|x+2|+3$ is shown below.

  

Hence,

Domain is $(-\infty ,\infty )$ , Range is $[3,\infty )$

The transformation of $g(x)=|x+2|+3$ is the $2$ units left shift and then $3$ units up shift of $f(x)=\left|x\right|$ .