Chapter 1.3, Problem 20E

Solution

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

The six functions that are increasing on their entire domains are:

  $\begin{array}{c}y=x,\\ y=x^3,\\ y=\sqrt{x},\\ y=e^x,\\ y=\ln x,\\ y=\frac{1}{1+e^{-x}}\end{array}$

Given information:

The six functions that are increasing on their entire domains.

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:

Whenever there is a positive change in $x$ values for any two points in the interval, the function is increasing.

From the given functions, the increasing functions are:

Thus, the increasing functions are:

  $y=x$

  

$y=x^3$

  

  $y=\sqrt{x}$

  

  $y=e^x$

  

$y=\ln x$

  

$y=\frac{1}{1+e^{-x}}$

  

Thus, the six increasing functions on their entire domains are:

  $\begin{array}{c}y=x,\\ y=x^3,\\ y=\sqrt{x},\\ y=e^x,\\ y=\ln x,\\ y=\frac{1}{1+e^{-x}}\end{array}$