Chapter 0.5, Problem 31E

To determine

To graph: The functions $f,f^{-1},y=x$ , using parametric graphing.

Explanation of Solution

Given information:

The given function is $f\left(x\right)={\sin }^{-1}x$ .

Method:

Transform the function $f\left(x\right)$ into parametric form by substituting suitable parameter and then find $f^{-1}\left(x\right)$ in parametric form.

Calculation:

The given function is $f\left(x\right)={\sin }^{-1}x$ . Let t be the parameter for the function $f\left(x\right)$ , that is, $x=t$ .

Then parametric equation for $f\left(x\right)$ is

  $x=t,y={\sin }^{-1}t$ … (1)

Now, find $f^{-1}\left(x\right)$ as follows:

If $y=f\left(x\right)$, therefore, the inverse function $f^{-1}\left(y\right)=x$.

As $y={\sin }^{-1}x$ , then$\begin{array}{c}\sin \left(y\right)=x\\ x=\sin \left(y\right)\end{array}$

Substitute $f^{-1}\left(y\right)$ for x ,

  $f^{-1}\left(y\right)=\sin \left(y\right)$

Change variable by replacing y by x

  $f^{-1}\left(x\right)=\sin \left(x\right)$

Bur $x=t$ , therefore, the parametric equation of $f^{-1}$ is

  $x=t,y=f^{-1}\left(t\right)\text{}\text{OR}y=\sin t$ … (2)

Parametric equation for $y=x$:

Choose parameter $x=t$ , then

  $x=t,y=t$

The parametric graphs of $f,f^{-1},y=x$ are shown in Fig. (1).

  

Interpretation:

The graph of the curves $f,f^{-1}$ are reflection of each other in the region $x=-1$ to $x=1$ .