Q. 86 - Precalculus Enhanced with Graphing Utilities

Question

In Problem $85 and 86$ graph each of the function

$g\left(x\right)=\left\{\begin{array}{l}2\sin x if 0\le x\le \mathrm{\pi }\\ \cos x+1 if\mathrm{\pi }<\mathrm{x}\le 2\mathrm{\pi }\end{array}\right.$

Step-by-Step Solution

Answer

Graph of the given function

Step 1.Given information

The given function $g\left(x\right)=\left\{\begin{array}{l}2\sin x if 0\le x\le \mathrm{\pi }\\ \cos x+1 if\mathrm{\pi }<\mathrm{x}\le 2\mathrm{\pi }\end{array}\right.$

Now graph each function as follows.
The given function is a composite function of function $y=2\sin x$ for the interval $0\le x\le \mathrm{\pi }$

and function $y=\cos x+1$ for the interval $\mathrm{\pi }<\mathrm{x}\le 2\mathrm{\pi }$.

Here domain of the function $g\left(x\right)$ is $[0,2\mathrm{\pi }]$ and its range is 1 to - 1.  

Step 2.Calculate some points for sin x in their given domain

Choose different x-values and calculate the corresponding $y=\sin x$ values.

x	y=2sinx	(x,y)
0	$2\sin 0=0$	$(0,0)$
$\frac{\mathrm{\pi }}{4}$	$$\begin{gathered} 2\sin \left(\frac{\mathrm{\pi }}{4}\right)=2\frac{\sqrt{2}}{2} \\ =\sqrt{2} \end{gathered}$$	$(\frac{\mathrm{\pi }}{4},\sqrt{2})$
$\frac{\mathrm{\pi }}{2}$	$$\begin{gathered} 2\sin \left(\frac{\mathrm{\pi }}{2}\right)=2\left(1\right) \\ =2 \end{gathered}$$	$(\frac{\mathrm{\pi }}{2},2)$
$\frac{3\mathrm{\pi }}{4}$	$$\begin{gathered} 2\sin \left(\frac{3\mathrm{\pi }}{4}\right)=2\frac{\sqrt{2}}{2} \\ =\sqrt{2} \end{gathered}$$	$\left(\frac{3\mathrm{\pi }}{4},\sqrt{2}\right)$
$\mathrm{\pi }$	$2\sin \left(\mathrm{\pi }\right)=0$	$\left(\mathrm{\pi },0\right)$

Step 3.Calculate some points for cos x in their given domain

Choose different -xvalues and calculate the $y=\cos x+1$ corresponding  values.

$x$	$y=\cos x+1$	$(x,y)$
$\frac{5\mathrm{\pi }}{4}$	$$\begin{gathered} \cos \left(\frac{5\mathrm{\pi }}{4}\right)+1=-\frac{\sqrt{2}}{2}+1 \\ =0.293 \end{gathered}$$	$\left(\frac{5\mathrm{\pi }}{4},0.293\right)$
$\frac{3\mathrm{\pi }}{2}$	$$\begin{gathered} \cos \left(\frac{3\mathrm{\pi }}{2}\right)+1=0+1 \\ =1 \end{gathered}$$	$\left(\frac{3\mathrm{\pi }}{2},1\right)$
$\frac{7\mathrm{\pi }}{4}$	$$\begin{gathered} \cos \left(\frac{7\mathrm{\pi }}{4}\right)=\frac{\sqrt{2}}{2}+1 \\ =1.7071 \end{gathered}$$	$\left(\frac{7\mathrm{\pi }}{4},1.7071\right)$
$2\mathrm{\pi }$	$$\begin{gathered} \cos \left(2\mathrm{\pi }\right)+1=1+1 \\ =2 \end{gathered}$$	$\left(2\mathrm{\pi },2\right)$

Step 4.Plot the above ordered pairs on the graph and connect them with a smooth curve.

Thus function $g\left(x\right)$ is graphed as a composite function of $y=2\sin x$ in blue and $y=\cos x+1$ in pink shown below: