Chapter 12.9, Problem 40PPS

a.

To determine

To graph the given function $y={\cos }^{-1}x$ . To describe its domain and range.

Answer to Problem 40PPS

Domain $\left[-1,1\right]$ & Range $\left[0,\pi \right]$

Explanation of Solution

Given:

A function is given as below:

  $y={\cos }^{-1}x$

  

The above figure shows the graph of function $y={\cos }^{-1}x$ .

The domain of the function is $\left[-1,1\right]$ and its range is $\left[0,\pi \right]$ .

b.

To determine

Write the function $y={\cos }^{-1}x$ with different notations.

Answer to Problem 40PPS

  $y=\mathrm{Arccos}x$

Explanation of Solution

Given: A function is given as below:

  $y={\cos }^{-1}x$

The given function can also be written as $y=\mathrm{Arccos}x$ .

c.

To determine

To evaluate this function to the nearest tenth by choosing a value for x between 0 and -1.

Answer to Problem 40PPS

for $x=\frac{1}{2}y=\frac{\pi }{3}$

Explanation of Solution

Given:

A function is given as below:

  $y={\cos }^{-1}x$

Calculation:

Let $x=\frac{1}{2}$ , Now

  $\begin{array}{l}y={\cos }^{-1}\frac{1}{2}\\ \Rightarrow \cos y=\frac{1}{2}\\ \Rightarrow y=\frac{\pi }{3}\end{array}$

d.

To determine

Compare the graphs of $y=\cos x\&y={\cos }^{-1}x$ .

Answer to Problem 40PPS

The domain of the $y={\cos }^{-1}x$ function is the range of the $y=\cos x$ function.

Explanation of Solution

The graph of $y={\cos }^{-1}x$ and $y=\cos x$ is shown below respectively,

  

Thus, we can see from the graphs that the domain of the first function is the range of the second function. But the range of the first function is not equal to the domain of the second function.