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Q. 86

Expert-verified
Found in: Page 411

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problem $85and86$ graph each of the function$g\left(x\right)=\left\{\begin{array}{l}2\mathrm{sin}xif0\le x\le \mathrm{\pi }\\ \mathrm{cos}x+1if\mathrm{\pi }<\mathrm{x}\le 2\mathrm{\pi }\end{array}\right\$

Graph of the given function

See the step by step solution

## Step 1.Given information

The given function $g\left(x\right)=\left\{\begin{array}{l}2\mathrm{sin}xif0\le x\le \mathrm{\pi }\\ \mathrm{cos}x+1if\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 $y=2\mathrm{sin}x$ for the interval $0\le x\le \mathrm{\pi }$

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

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

## Step 2.Calculate some points for sinx in their given domain

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

 x y=2sinx (x,y) 0 $2\mathrm{sin}0=0$ $\left(0,0\right)$ $\frac{\mathrm{\pi }}{4}$ $2\mathrm{sin}\left(\frac{\mathrm{\pi }}{4}\right)=2\frac{\sqrt{2}}{2}\phantom{\rule{0ex}{0ex}}=\sqrt{2}$ $\left(\frac{\mathrm{\pi }}{4},\sqrt{2}\right)$ $\frac{\mathrm{\pi }}{2}$ $2\mathrm{sin}\left(\frac{\mathrm{\pi }}{2}\right)=2\left(1\right)\phantom{\rule{0ex}{0ex}}=2$ $\left(\frac{\mathrm{\pi }}{2},2\right)$ $\frac{3\mathrm{\pi }}{4}$ $2\mathrm{sin}\left(\frac{3\mathrm{\pi }}{4}\right)=2\frac{\sqrt{2}}{2}\phantom{\rule{0ex}{0ex}}=\sqrt{2}$ $\left(\frac{3\mathrm{\pi }}{4},\sqrt{2}\right)$ $\mathrm{\pi }$ $2\mathrm{sin}\left(\mathrm{\pi }\right)=0$ $\left(\mathrm{\pi },0\right)$

## Step 3.Calculate some points for cosx in their given domain

Choose different -xvalues and calculate the $y=\mathrm{cos}x+1$ corresponding values.
 $x$ $y=\mathrm{cos}x+1$ $\left(x,y\right)$ $\frac{5\mathrm{\pi }}{4}$ $\mathrm{cos}\left(\frac{5\mathrm{\pi }}{4}\right)+1=-\frac{\sqrt{2}}{2}+1\phantom{\rule{0ex}{0ex}}=0.293$ $\left(\frac{5\mathrm{\pi }}{4},0.293\right)$ role="math" localid="1646328747587" $\frac{3\mathrm{\pi }}{2}$ $\mathrm{cos}\left(\frac{3\mathrm{\pi }}{2}\right)+1=0+1\phantom{\rule{0ex}{0ex}}=1$ $\left(\frac{3\mathrm{\pi }}{2},1\right)$ $\frac{7\mathrm{\pi }}{4}$ $\mathrm{cos}\left(\frac{7\mathrm{\pi }}{4}\right)=\frac{\sqrt{2}}{2}+1\phantom{\rule{0ex}{0ex}}=1.7071$ $\left(\frac{7\mathrm{\pi }}{4},1.7071\right)$ $2\mathrm{\pi }$ $\mathrm{cos}\left(2\mathrm{\pi }\right)+1=1+1\phantom{\rule{0ex}{0ex}}=2$ $\left(2\mathrm{\pi },2\right)$

## 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 $y=2\mathrm{sin}x$ in blue and $y=\mathrm{cos}x+1$ in pink shown below: