Chapter 1.3, Problem 26E

Solution

To Find: The suitable functions from twelve basic functions based on the given description.

The three functions with end behaviour s are:

  $\begin{array}{c}y=x\\ y=x^3\\ y=\mathrm{int}x\end{array}$

Given information:

The three functions with end behaviour $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$ .

Given functions are:

  $\begin{array}{l}1.f\left(x\right)=x2.f\left(x\right)=x^2\\ 3.f\left(x\right)=x^{3}4.f\left(x\right)=\frac{1}{x}\\ 5.f\left(x\right)=\sqrt{x}6.f\left(x\right)=e^x\\ 7.f\left(x\right)=\ln x8.f\left(x\right)=\sin x\\ 9.f\left(x\right)=\cos x10.f\left(x\right)=\left|x\right|\\ 11.f\left(x\right)=\mathrm{int}x12.f\left(x\right)=\frac{1}{1+e^{-x}}\end{array}$

Calculation:

As a function goes towards either end of the x-axis, a horizontal asymptote provides one kind of end behaviour.

The three functions with end behaviour are $y=x,y=x^3,y=\mathrm{int}x$ .

The graphical representation of the functions $y=x,y=x^3,y=\mathrm{int}x$ is as follows:

  

From the figure, it is clear that the graphs of $y=x,y=x^3,y=intx$ have end behaviour because it satisfies the ideology of end behaviour and also when $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty$ .

The functions are also defined, when substitute the value of $x=-\infty$ .

Thus, the three functions with end behaviour $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$ are:

  $\begin{array}{c}y=x\\ y=x^3\\ y=\mathrm{int}x\end{array}$