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.

$y=\cos \left(-2x\right)$

The function to be plotted is :

$$\begin{gathered} y=\cos (-2x) \\ y=\cos 2x \end{gathered}$$

since cosine is an even function.

By comparing the given function$y=\cos \left(2x\right)$

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

we get amplitude: $A=1$

Time period :$$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega } \\ T=\frac{2\mathrm{\pi }}{2} \\ T=\mathrm{\pi } \end{gathered}$$

The graph will lie between -1 and 1 on the y-axis. One cycle begins at $x=0$and ends at$x=\mathrm{\pi }$.

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

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

The x-coordinates of the five key points are :

First x-coordinate is: 0

second x-coordinate is: role="math" localid="1646927120565" $0+\frac{\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{4}$

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

Fourth x-coordinate is: $\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{4}=\frac{3\mathrm{\pi }}{4}$

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

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

Since $y=\cos \left(2x\right)$

Hence multiply the y-coordinates of the five key points for $y=\cos x$ by 1

The five key points on the graph are:

$\left(0,1\right),\left(\frac{\mathrm{\pi }}{4},0\right),\left(\frac{\mathrm{\pi }}{2},-1\right),\left(\frac{3\mathrm{\pi }}{4},0\right),\left(\mathrm{\pi },1\right)$

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 $\frac{\mathrm{\pi }}{4}$radian.

As we can see that the value of $x$is a 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 the range of the function is$\left[-1,1\right]$