Chapter 3.5, Problem 23E

To determine

To find: the end behavior of the given function.

Answer to Problem 23E

  $\begin{array}{l}g\left(x\right)\to \infty \text{ as }x\to \infty \\ g\left(x\right)\to \infty \text{ as }x\to -\infty \end{array}$

Explanation of Solution

Given information:

  $g\left(x\right)=\left|{\left(x-3\right)}^2-1\right|$

Calculation:

Let $g\left(x\right)=y$ . Make a table for $x$ and y as follows:

$x$	$y$
$-10000$	$1.0006\times {10}^8$
$-1000$	$1.006\times {10}^6$
$-100$	$10608$
0	8
$100$	9408
$1000$	994008
$10000$	$9.994001\times {10}^7$

Since the function is an absolute value, so the function will be infinity as x approaches to infinity or negative infinity. Thus, the end behavior is $g\left(x\right)\to \infty \text{ as }x\to \infty$ and $g\left(x\right)\to \infty \text{ as }x\to -\infty$ .