Chapter 5, Problem 54RE

(a)

To determine

To find: TheGraphing device to graph the function.

Answer to Problem 54RE

Through graph range can determined which is

  $\text{Range=}\left[\text{-0}\text{.841,0}\text{.841}\right]$ .

Explanation of Solution

Given: $\text{y=Sin}\left(\text{Cosx}\right)$

Concept used:

Modulus function make the graph shifted in the region of positive.

  $\text{Cosx}$ graph stared from 1.

Its $\text{Domain=}\left(-\infty ,\infty \right)$ and $\text{Range=}\left[\text{0,1}\right]$

Calculation:

Thee graph of the $\text{y=Sin}\left(\text{Cosx}\right)$ .

  

Graphing device use here is Desmos graphing calculator. from the graph

Through graph range can determined which is

  $\text{Range=}\left[\text{-0}\text{.841,0}\text{.841}\right]$ .

From the graph it can be easily verified the period here is $\text{π}$ .

(b)

To determine

To find:whether the Function is periodic if then described it.

Answer to Problem 54RE

The period here is $\text{π}\text{.}$

Explanation of Solution

Given: $\text{y=Sin}\left(\text{Cosx}\right)$

Concept used:

Period is measured as distance it takes for the entire graph to repeat.

Since the period of the Cosine function is $\text{π}$ .

Calculation:

  $\text{y=Sin}\left(\text{Cosx}\right)$

Form the graph its clearly verified the periods as:

Form its range:

  $\text{Periods=}\left(\frac{\text{π}}{\text{2}}\text{-}\left(\text{-}\frac{\text{π}}{\text{2}}\right)\right)\text{=π}\text{.}$

Hence the period here is $\text{π}\text{.}$

(c)

To determine

To find: The graph whether the function is even or odd.

Answer to Problem 54RE

The given function is Odd.

Explanation of Solution

Given: $\text{y=Sin}\left(\text{Cosx}\right)$

Concept used:

  $\text{f}\left(\text{-x}\right)\text{=f}\left(\text{x}\right)$ the function is an even function.

  $\text{f}\left(\text{-x}\right)\text{=-f}\left(\text{x}\right)$ the function is and odd function

Calculation:

  $\text{y=Sin}\left(\text{Cosx}\right)$

Since, $\text{Cosx and }\left|\text{x}\right|$ are even functions therefore its composition $y=\left|\mathrm{Cos}x\right|$ is also an even function.

  $f\left(-x\right)=\text{y=Sin}\left(\text{Cos}\left(\text{-x}\right)\right)=\text{y=Sin}\left(\text{Cos}\left(\text{x}\right)\right)=-f\left(x\right)\forall x.$

Her the function

  $\text{Cosx function but Sinx is odd}\text{.}$

Hence the given function is Odd.