n 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

$y=-3\cos x$

The function to be plotted is

$y=-\cos x$

By comparing the given function$y=-3\cos x$

with$y=A\cos \left(\omega x\right)$

we get amplitude:$\left|A\right|=3$

time period

$$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega } \\ T=\frac{2\mathrm{\pi }}{1} \\ T=2\mathrm{\pi } \end{gathered}$$

The graph will lie between -3 and 3 on the y-axis. One cycle begins at $0$and ends at$2\mathrm{\pi }$

Divide the interval $2\mathrm{\pi }$into four subintervals,

each of length

$\frac{2\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{2}$

The x-coordinates of the five key points are :

First x-coordinate is 0

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

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

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

Fifth x-coordinate is$\frac{3\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}=2\mathrm{\pi }$

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

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 $\frac{\mathrm{\pi }}{2}$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 -3 to 3

So range of the function is$\left[-3,3\right]$