Chapter 2.2, Problem 51E

Solution

To write: Discuss end behaviour of a function.

The end behaviour of the graph of functions $f\left(x\right)$ and $g\left(x\right)$ are: (i) $f\left(x\right)\to +\infty$ , as $x\to -\infty$ and $f\left(x\right)\to -\infty$ as $x\to \infty$ , (ii) $g\left(x\right)\to +\infty$ , as $x\to +\infty$ and $f\left(x\right)\to -\infty$ as $x\to -\infty$ .

Given data: The end behaviour of the function $f\left(x\right)$ is such that $f\left(x\right)\to +\infty$ as $x\to -\infty$ and $f\left(x\right)\to -\infty$ as $x\to \infty$ and another function $g\left(x\right)$ whose end behaviour is to discuss is given by $g\left(x\right)=-f\left(x\right)$ .

Method/Formula used:

An expression of the form $f\left(x\right)=\pm a_0\pm a_{1}x\pm a_2{x}^2\pm \cdots \pm a_n{x}^n$ is called the polynomial of degree n if and only if,

(i) $a_0,a_1,a_2,\dots ,a_n$ , are real numbers

(ii) n is a non-negative integer.

Calculation:

A function (polynomial) of degree n can be written as: $f\left(x\right)=\pm a_0\pm a_n{x}^n$

Choose $a_0=2$ , $a_n=1$ , and $n=3$ .

Therefore, the required polynomial is $f\left(x\right)=\pm 2\pm 1x^3$ , so there are four different polynomials represented by the function $f\left(x\right)$ .

Consider the polynomial function $f\left(x\right)=2-x^3$ , then

(i) If $x\to -\infty$ ,

  $\begin{array}{c}f\left(-\infty \right)=2-{\left(-\infty \right)}^3\because {\left(-\infty \right)}^{\text{odd}\text{number}}=-\infty \\ =2-\left(-\infty \right)\\ =2+\infty \because \infty +a\left(\text{numbe}r\right)=-\infty \\ =+\infty \end{array}$

Thus, $f\left(x\right)\to +\infty$ , when $x\to -\infty$ .

(i) If $x\to \infty$ ,

  $\begin{array}{c}f\left(\infty \right)=2-{\left(\infty \right)}^3\because {\left(\infty \right)}^{\text{odd}\text{number}}=\infty \\ =3-\infty \\ =-\infty \end{array}$

  $f\left(x\right)\to -\infty$ , when $x\to +\infty .$

Now construct a function $g\left(x\right)$ such that $g\left(x\right)=-f\left(x\right)$ . Thus, the function $g\left(x\right)$ is

  $\begin{array}{c}g\left(x\right)=-f\left(x\right)\\ =-\left(2-x^3\right)\\ =x^3-2\end{array}$

The graphs of functions $g\left(x\right)$ and $f\left(x\right)$

in the same window are shown in Fig. (1):

  

Interpretation: As clear from the graph the end behaviour of the graph of functions $f\left(x\right)$ and $g\left(x\right)$ are:

(i) $f\left(x\right)\to +\infty$ , as $x\to -\infty$ and $f\left(x\right)\to -\infty$ as $x\to \infty$

(ii) $g\left(x\right)\to +\infty$ , as $x\to +\infty$ and $f\left(x\right)\to -\infty$ as $x\to -\infty$

Thus, the behaviours of functions $f\left(x\right)$ and $g\left(x\right)$ are as if they are additive inverse of each other’s.

This graph also defines the additive inverse of a function.