91Ó°ÊÓ

We have been given a function.

We have to graph this function and determine the domain and the range of the function using the graph.

On comparing, we get:

Amplitude $A=\frac{3}{2}$

Time period $T=\frac{2\mathrm{Ï€}}{\mathrm{Ó\neg }}$

$$\begin{gathered} \mathrm{Ó\neg }=\frac{\mathrm{Ï€}}{4} \\ T=8 \end{gathered}$$

The graph lies between $\frac{3}{2}$and $-\frac{3}{2}$.

One cycle starts from $x=0$and ends at $x=8$.

Five key points have x-coordinates as:

$0,2,4,6,8$

For the second cycles, x-coordinates of key points will be:

$10,12,14,16$

x and y coordinates for the first cycles are:

role="math" localid="1646397454423" $(0,-\frac{3}{2}),(2,0),(4,\frac{3}{2}),(6,0),(8,-\frac{3}{2})$

x and y coordinates for the second cycles are:

role="math" localid="1646397509167" $(10,0),(12,\frac{3}{2}),(14,0),(16,-\frac{3}{2})$

x and y coordinates for the first cycles will become:

$(0,-1),(2,\frac{1}{2}),(4,2),(6,\frac{1}{2}),(8,-1)$

x and y coordinates for the second cycles will become:

$(10,\frac{1}{2}),(12,2),(14,\frac{1}{2}),(16,-1)$

The graph continues indefinitely to the left and to the right, the domain is the set of all real numbers. Therefore, the domain is $\left\{x\right|-∞<x<∞\}$.

The range of the function consists of all real numbers from $-1$to $2$, inclusive.

Therefore, the range is $\left\{y\right|-1≤y≤2\}$.