Chapter 1.3, Problem 27E

Solution

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

The four graphs look the same even when flipped upside down are:

  $\begin{array}{l}y=x\\ y=x^3\\ y=\frac{1}{x}\\ y=\sin x\end{array}$

Given information:

The four functions whose graphs look the same when turned upside-down and flipped about the $y-$ axis.

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:

By turned upside-down, it has been understood that a function has been flipped about the $x-$ axis (which means setting $y=-y$ ). Then flip about the $y-$ axis by setting $x=-x$ . So, for example,

  $y=f\left(x\right)$

Flip the function about the $x-$ axis:

  $-y=f\left(x\right)$

And the $y-$ axis:

  $\begin{array}{c}-y=f\left(-x\right)\\ y=-f\left(-x\right)\end{array}$

It is required that $y$ to be the same for all $x,$ then

  $\begin{array}{c}f\left(x\right)=-f\left(-x\right)\\ -f\left(x\right)=f\left(-x\right)\end{array}$

Likewise, in another function $f\left(x\right)=\sin x$ , replace $x$ by $-x$ :

  $\begin{array}{c}f\left(-x\right)=\sin \left(-x\right)\\ f\left(-x\right)=-\sin x\\ f\left(-x\right)=-f\left(x\right)\end{array}$

Also, in function

The graph of $y=x,y=x^3,y=\frac{1}{x},y=\sin x$

on the same viewing window is shown below:

  

Thus, the four graphs look the same even when flipped upside down are:

  $\begin{array}{l}y=x\\ y=x^3\\ y=\frac{1}{x}\\ y=\sin x\end{array}$