Chapter 2.1, Problem 6DC

Solution

To make a conjecture about the end behaviour of the function $f\left(x\right)=x^6-x$ .

Analysing the graph of the function $f\left(x\right)=x^6-x$ , it is observed that $f\left(x\right)\to \infty$ as $x\to \infty$ and $x\to -\infty$ .

Thus, a conjecture is made on the end behaviour of $f\left(x\right)=x^6-x$ :

  $f\left(x\right)=x^6-x$ approaches $\infty$ as $x\to \infty$ and $x\to -\infty$ .

Given the function:

  $f\left(x\right)=x^6-x$

Calculation:

Graph the function $f\left(x\right)=x^6-x$:

  

It seems like $f\left(x\right)$ is approaching $\infty$ as $x$ progresses to $\infty$ on either sides of the $x$ -axis.

So it is reasonable to make the conjecture that $f\left(x\right)=x^6-x$ approaches $\infty$ as $x\to \infty$ and $x\to -\infty$ .

Thus, a conjecture is made on the end behaviour of $f\left(x\right)=x^6-x$ :

  $f\left(x\right)=x^6-x$ approaches $\infty$ as $x\to \infty$ and $x\to -\infty$ .

Conclusion:

Analysing the graph of the function $f\left(x\right)=x^6-x$ , it is observed that $f\left(x\right)\to \infty$ as $x\to \infty$ and $x\to -\infty$ .

Thus, a conjecture is made on the end behaviour of $f\left(x\right)=x^6-x$ :

  $f\left(x\right)=x^6-x$ approaches $\infty$ as $x\to \infty$ and $x\to -\infty$ .