Chapter 1.3, Problem 23E

Solution

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

The three functions with no zeros are:

  $\begin{array}{c}y=\frac{1}{x}\\ y=e^x\\ y=\frac{1}{1+e^{-x}}\end{array}$

Given information:

The three functions with no zeros.

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:

The graph of the three functions on the same viewing window is shown below:

  

There are no zeros in each of the three functions, as can be seen from the graph.

It is also important to note that none of these functions pass through the origin.

Thus, the three functions with no zeros.

  $\begin{array}{c}y=\frac{1}{x}\\ y=e^x\\ y=\frac{1}{1+e^{-x}}\end{array}$