Chapter 4.7, Problem 82E

To determine

To graph: For the function and compare the graph of $g$ with the graph of $f\left(x\right)=\arcsin x$ .

Explanation of Solution

Given information:

The given function is $g\left(x\right)=\arcsin \frac{x}{2}$ .

Graph:

The domain in the function $g\left(x\right)=\arcsin \frac{x}{2}$ .

In case of the function $f\left(x\right)=\arcsin x$ the domain of the function is $-1\le x\le 1$ .

And the range of the function $f\left(x\right)=\arcsin x$ is $-\frac{\pi }{2}\le f\left(x\right)\le \frac{\pi }{2}$ .

In case of the function $g\left(x\right)=\arcsin \frac{x}{2}$ the domain of the function is $-1\le \frac{x}{2}\le 1$ .

Using above interval.

  $\begin{array}{c}-1\left(2\right)\le x\le 1\left(2\right)\\ -2\le x\le 2\end{array}$

And the range of the function $g\left(x\right)=\arcsin \frac{x}{2}$ remains same that is $-\frac{\pi }{2}\le g\left(x\right)\le \frac{\pi }{2}$ .

The graph of the functions $g\left(x\right)=\arcsin \frac{x}{2}$ and $f\left(x\right)=\arcsin x$ is shown in figure (1).

  

Figure (1)

The difference between the graph of $f\left(x\right)=\arcsin x$ and $g\left(x\right)=\arcsin \frac{x}{2}$ is.

The domain of the function $f\left(x\right)=\arcsin x$ is in interval $-1\le x\le 1$ .

The domain of the function $g\left(x\right)=\arcsin \frac{x}{2}$ is in interval $-2\le x\le 2$ .

Interpretation:The domain of the function $f\left(x\right)=\arcsin x$ is in interval $-1\le x\le 1$ and the domain of the function $g\left(x\right)=\arcsin \frac{x}{2}$ is in interval $-2\le x\le 2$ .