Chapter 4.3, Problem 30E

a

To determine

To draw: The graph of the given function.

Explanation of Solution

Given information: Given function $f(x)={\sin }^{-1}(\sin x)$

Graph:

  

The given graph is in the viewing window of $[-2\pi ,2\pi ]\text{by}[-4,4]$

(b)

To determine

To Find: The domain of the given function.

Answer to Problem 30E

Domain is $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$

Explanation of Solution

Given information: Given function $f(x)={\sin }^{-1}(\sin x)$

The domain of ${\sin }^{-1}$ is $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$ which can be seen from the above graph clearly.

(c)

To determine

To Find: The range of the given function.

Answer to Problem 30E

Range is $\left[-1,1\right]$

Explanation of Solution

Given information: Given function $f(x)={\sin }^{-1}(\sin x)$

The range of ${\sin }^{-1}$ is $\left[-1,1\right]$ which can be seen from the above graph clearly.

(d)

To determine

To state: The point where the given function is not differentiable,

Answer to Problem 30E

The given function is differentiable for all values of x.

Explanation of Solution

Given information: Given function $f(x)={\sin }^{-1}(\sin x)$

From the graph it can be seen that the given function is continuous, hence it is differentiable at all values of x .

(e)

To determine

To draw: The graph of the derivative of the given function.

Explanation of Solution

Given information: Given function $f(x)={\sin }^{-1}(\sin x)$

Graph:

  

From the graph we can see that $f\text{'}(x)=1$

(f)

To determine

To Find: The derivative of the given function.

Answer to Problem 30E

The derivative is 1.

Explanation of Solution

Given information: Given function $f(x)={\sin }^{-1}(\sin x)$ .

Formula Used: ${\sin }^{-1}(\sin x)=x\text{and}\frac{d(x)}{dx}=1$

Calculation:

  $\begin{array}{l}f(x)={\sin }^{-1}(\sin x)\\ f(x)=x\\ \text{Differentiation}\text{with}\text{respect}\text{to}x\\ f\text{'}(x)=\frac{d(x)}{dx}\\ f\text{'}(x)=1\end{array}$