Chapter 1.3, Problem 24E

Solution

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

The three functions with range $\text{{all}\text{real}\text{numbers}}$ are:

  $\begin{array}{c}y=x,\\ y=x^3\\ y=\ln x\end{array}$

Given information:

The three functions with range $\text{{all}\text{real}\text{numbers}}$ .

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:

A real number can be expressed as a function defined by the identity function, the cubing function, and the natural logarithm function. Those functions are $y=x,y=x^3$ and $y=\ln x$ .

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

  

It can be clearly seen that the graph of functions $y=x,y=x^3$ and $y=\ln x$ shows the range which consists only of real numbers.

Thus, the three functions with range $\text{{all}\text{real}\text{numbers}}$ are:

  $\begin{array}{c}y=x,\\ y=x^3\\ y=\ln x\end{array}$