Chapter 1.3, Problem 21E

Solution

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

The three functions that are decreasing on the interval $\left(-\infty ,0\right)$ are:

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

Given information:

The three functions that are decreasing on the interval $\left(-\infty ,0\right)$ .

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 three functions that are decreasing on the interval $\left(-\infty ,0\right)$ are:

  $y=-e^x$

  

$y=\frac{1}{x}$

  

  $y=\left|x\right|$

  

Thus, the three functions that are decreasing on the interval $\left(-\infty ,0\right)$ are:

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