Chapter 1, Problem 6P

To determine

To sketch: The function $g\left(x\right)=\left|x^2-1\right|-\left|x^2-4\right|$.

Explanation of Solution

Given:

The function $g\left(x\right)=\left|x^2-1\right|-\left|x^2-4\right|$.

Result used:

Even function:

A function is even if and only if $g\left(x\right)=g\left(-x\right)$ for all x in the domain of g.

Definition used:

The absolute value of x is defined as, $\left|x\right|=\{\begin{array}{l}x\text{if }x\ge 0\\ -x\text{if }x<0\end{array}$.

Calculation:

Consider the function $g\left(x\right)=\left|x^2-1\right|-\left|x^2-4\right|$

Now, $\left|x^2-1\right|=\{\begin{array}{l}{x}^2-1\text{if}x\ge 1\\ 1-x^2\text{if}x<1\end{array}$

$\left|x^2-4\right|=\{\begin{array}{l}{x}^2-4\text{if}x\ge 2\\ 4-x^2\text{if}x<2\end{array}$

The function can be split into different cases as follows:

Case (i):

For $0\le x\le 1$

$\begin{array}{l}{g}_1\left(x\right)=x^2+1-x^2-4\\ g_1\left(x\right)=-3\end{array}$

To sketch: Use the online graphing calculator and draw the graph of the function: $g_1\left(x\right)=-3$ as shown below in Figure 1.

Case (ii):

For $1\le x<2$

$\begin{array}{l}{g}_2\left(x\right)=x^2-1+x^2-4\\ g_2\left(x\right)=2x^2-5\end{array}$

To sketch: Use the online graphing calculator and draw the graph of the function: $g_2\left(x\right)=2x^2+5$ as shown below in Figure 2

Case (iii):

For $x\ge 2$

$\begin{array}{l}{g}_3\left(x\right)=x^2-1-x^2+4\\ g_3\left(x\right)=3\end{array}$

To sketch: Use the online graphing calculator and draw the graph of the function: $g_3\left(x\right)=3$ as shown below in Figure 3.

Case (iv):

For $-1<x\le 0$

$\begin{array}{l}{g}_1\left(-x\right)=x^2+1-x^2-4\\ g_1\left(-x\right)=-3\end{array}$

To sketch: Use the online graphing calculator and draw the graph of the function: $g_1\left(-x\right)=-3$ as shown below in Figure 4.

Case (v):

For $-2<x\le -1$

$\begin{array}{l}{g}_2(-x)=x^2-1+x^2-4\\ g_2(-x)=2x^2-5\end{array}$

To sketch: Use the online graphing calculator and draw the graph of the function: $g_2\left(-x\right)=2x^2+5$ as shown below in Figure 5.

Case (vi):

For $x\le -2$

$\begin{array}{l}{g}_3\left(-x\right)=x^2-1+x^2-4\\ g_3\left(-x\right)=3\end{array}$

To sketch: Use the online graphing calculator and draw the graph of the function: $g_3\left(-x\right)=3$ as shown below in Figure 6.

Combine all the graphs in figure 1, 2,3,4,5 and 6 as shown below in Figure 7.

To sketch: Use the online graphing calculator and draw the graph of the function: $g\left(x\right)=\left|x^2-1\right|-\left|x^2-4\right|$ as shown below in Figure 7.