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.

$y=2\cos \left(\frac{1}{4}x\right)$

The function to be plotted is :

$y=2\cos \left(\frac{1}{4}x\right)$

Divide the interval $\left[0,8\mathrm{\pi }\right]$into four subintervals,

each of length:$\frac{8\mathrm{\pi }}{4}=2\mathrm{\pi }$

The x-coordinates of the five key points are :

First x-coordinate is 0

second x-coordinate is$0+2\mathrm{\pi }=2\mathrm{\pi }$

Third x-coordinate is$2\mathrm{\pi }+2\mathrm{\pi }=4\mathrm{\pi }$

Fourth x-coordinate is $4\mathrm{\pi }+2\mathrm{\pi }=6\mathrm{\pi }$

Fifth x-coordinate is$6\mathrm{\pi }+2\mathrm{\pi }=8\mathrm{\pi }$

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

Since $y=2\cos \left(\frac{1}{4}x\right)$

Multiply the y-coordinates of the five key points for $y=\cos x$by 2.

The five key points on the graph are :

role="math" localid="1646929287738" $\left(0,2\right),\left(2\mathrm{\pi },0\right),\left(4\mathrm{\pi },-2\right),\left(6\mathrm{\pi },0\right),\left(8\mathrm{\pi },2\right)$

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 $2\mathrm{\pi }$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 -1 to 1

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