Chapter 3.8, Problem 50PPS

a.

To determine

To complete the given table for the function $f(x)=\left|〚x〛\right|$ and $g(x)=〚\left|x\right|〛$ .

Explanation of Solution

Given:

Function: $f(x)=\left|〚x〛\right|$ and $g(x)=〚\left|x\right|〛$

Calculation:

Table of values for $f(x)=\left|〚x〛\right|$ and $g(x)=〚\left|x\right|〛$ :

$x$	$〚x〛$	$f(x)=\left|〚x〛\right|$	$\left|x\right|$	$g(x)=〚\left|x\right|〛$
$-3$	$-3$	$3$	$3$	$3$
$-25$	$-25$	$25$	$25$	$25$
$-2$	$-2$	$2$	$2$	$2$
$0$	$0$	$0$	$0$	$0$
$0.5$	$0$	$0$	$0.5$	$0$
$1$	$1$	$1$	$1$	$1$
$1.5$	$1$	$1$	$1.5$	$1$

b.

To determine

To graph $f(x)=\left|〚x〛\right|$ and $g(x)=〚\left|x\right|〛$ .

Explanation of Solution

Given:

Function: $f(x)=\left|〚x〛\right|$ and $g(x)=〚\left|x\right|〛$

Concept used:

The general form of an absolute value function is $f(x)=a\left|x-h\right|+k$ .

The graph of $f(x)=a\left|x-h\right|+k$ is a “V” with vertex $\left(h,k\right)$ , slope $m=a$ on the right side of the vertex $\left(x>h\right)$ and slope $m=-a$ on the left side of the vertex $\left(x<h\right)$ .

The graph of $f(x)=-a\left|x-h\right|+k$ is an upside-down “V” with vertex $\left(h,k\right)$ , slope $m=-a$ for $\left(x>h\right)$ and slope $m=a$ for $\left(x<h\right)$ .

Calculation for graph:

Consider $f(x)=\left|〚x〛\right|$

Compare with the general form:

  $f(x)=a\left|x-h\right|+k$

Here, the value of $a=1$ , so the graph opens upwards with a slope of $1$ .

The value of $h=0$ and the value of $k=0$ , so the vertex of the graph is $\left(0,0\right)$ .

By taking different values of x as taken in the table in (a) part, the graph can be plotted.

Graph:

The graph of $f(x)=\left|〚x〛\right|$ :

  

Now, consider $g(x)=〚\left|x\right|〛$

Compare with the general form:

  $g(x)=a\left|x-h\right|+k$

Here, the value of $a=1$ , so the graph opens upwards with a slope of $1$ .

The value of $h=0$ and the value of $k=0$ , so the vertex of the graph is $\left(0,0\right)$ .

By taking different values of x as taken in the table in (a) part, the graph can be plotted.

Graph:

The graph of $g(x)=〚\left|x\right|〛$ :

  

c.

To determine

To compare the graphs of $f(x)=\left|〚x〛\right|$ and $g(x)=〚\left|x\right|〛$ .

Explanation of Solution

Function: $f(x)=\left|〚x〛\right|$ and $g(x)=〚\left|x\right|〛$

From the above graphs, it is clear that, the graphs of both the functions are same.