In Problems 35–58, graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function.

$y=2\sin \left(\frac{1}{2}x\right)$

The function to be plotted is:

$y=2\sin \left(\frac{1}{2}x\right)$

Divide the interval $\left[0,4\mathrm{\pi }\right]$into four subintervals,

each of length:$\frac{4\mathrm{\pi }}{4}=\mathrm{\pi }$

The x-coordinates of the five key points are :

The first x-coordinate is 0

second x-coordinate is:$0+\mathrm{\pi }=\mathrm{\pi }$

Third x-coordinate is:$\mathrm{\pi }+\mathrm{\pi }=2\mathrm{\pi }$

Fourth x-coordinate is: $2\mathrm{\pi }+\mathrm{\pi }=3\mathrm{\pi }$

Fifth x-coordinate is:$3\mathrm{\pi }+\mathrm{\pi }=4\mathrm{\pi }$

These values represent the x-coordinates of the five key points on the graph.

Since $y=2\sin \left(\frac{1}{2}x\right)$

Hence, multiply the y-coordinates of the five key points for $y=\sin x$ by 2

The five key points on the graph are

$\left(0,0\right),\left(\mathrm{\pi },2\right),\left(2\mathrm{\pi },0\right),\left(3\mathrm{\pi },2\right),\left(4\mathrm{\pi },0\right)$.

Plot the five key points obtained in Step 4 and fill in the graph. Extend the graph in each direction to obtain the complete graph. Notice that additional key points appear every $\mathrm{\pi }$radian.

As we can see that the value of $x$is set of all real number.

So domain is $\left(-\infty ,\infty \right)$

The y- value of the function in the graph lies from -2 to 2

So the range of the function is$\left[-2,2\right]$