Chapter 5, Problem 56RE

(a)

To determine

To find: TheGraphing device to graph the function.

Answer to Problem 56RE

The function has $\text{Domain=}\left(-\infty ,\infty \right)$ and $\text{Range=}\left[\text{1}\text{.5, 3}\right]$ .

Explanation of Solution

Given: ${\text{y=1+2}}^{\text{cosx}}$

Concept used:

  $\text{Cosx}$ is even function over all equation change to $\text{even}$ .

Its $\text{Domain=}\left(-\infty ,\infty \right)$ and $\text{Range=}\left[\text{1}\text{.5, 3}\right]$ .

Amplitude is $\text{A=}\left(3-1.5\right)=1.5$

Calculation:

Thee graph of the ${\text{y=1+2}}^{\text{cosx}}$ .

  

Graphing device use here is Desmos graphing calculator. from the graph modulus symbolize the positivity of the function which is shown in the graph.

Through graph range and domain can Measured which is

  $\text{Domain=}\left(-\infty ,\infty \right)$ and $\text{Range=}\left[\text{1}\text{.5, 3}\right]$ .

Hence the function has $\text{Domain=}\left(-\infty ,\infty \right)$ and $\text{Range=}\left[\text{1}\text{.5, 3}\right]$ .

(b)

To determine

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

Answer to Problem 56RE

The period here is $\text{2π}$ .

Explanation of Solution

Given: ${\text{y=1+2}}^{\text{cosx}}$

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:

  ${\text{y=1+2}}^{\text{cosx}}$

Form the graph its clearly verified the periods as:

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

Hence the period here is $\text{2π}$ .

(c)

To determine

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

Answer to Problem 56RE

The given function is even.

Explanation of Solution

Given: ${\text{y=1+2}}^{\text{cosx}}$

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:

  ${\text{y=1+2}}^{\text{cosx}}$

Since $\mathrm{Cos}x$ even functions and overall becomes even function so, the function is even function.

  $\text{f}\left(\text{-x}\right){\text{=1+2}}^{\text{cos}\left(\text{-x}\right)}{\text{=1+2}}^{\text{cosx}}\text{=f}\left(\text{x}\right)\forall x.$

It looks like infinitely many arches of length $\text{π}$ and of height 1 in the upper half plane touching the x-axis and symmetrical about y-axis. Therefore, the function takes the same value at points on the x-axis which are equidistant form the y-axis.

Hence the given function is even.