Chapter 4, Problem 11CT

To determine

Tograph:The function $g\left(x\right)=-2\sin \left(\frac{\pi }{4}-x\right)$ including two full periods.

Explanation of Solution

Given information:

The given function is $g\left(x\right)=-2\sin \left(\frac{\pi }{4}-x\right)$

Formulaused:

The period of function $f\left(x\right)$ is T if $f\left(x\pm T\right)=f\left(x\right)$ .

Calculations:

The given function is $g\left(x\right)=-2\sin \left(\frac{\pi }{4}-x\right)$ . If T is the period of $g\left(x\right)$ then $g\left(x\pm T\right)=g\left(x\right)$ .

  $\Rightarrow -2\sin \left(\frac{\pi }{4}-x\pm T\right)=-2\sin \left(\frac{\pi }{4}-x\right)$

  $\Rightarrow T=2\pi$

  $\because \sin \left(x+2\pi \right)=\sin x$

  $g\left(x\pm 2\pi \right)=-2\sin \left(\frac{\pi }{4}-(x\pm 2\pi )\right)$

  $\Rightarrow g\left(x\pm 2\pi \right)=-2\sin \left(\frac{\pi }{4}-2\pi -x\right),-2\sin \left(\frac{\pi }{4}+2\pi -x\right)$

  $\Rightarrow g\left(x\pm 2\pi \right)=-2\sin \left(-\frac{7\pi }{4}-x\right),-2\sin \left(\frac{9\pi }{4}-x\right)$ .

The graph of function $g\left(x\right)=-2\sin \left(\frac{\pi }{4}-x\right)$ including two full periods is shown in Figure-1.

  

The periods of the function $g\left(x\right)$ are $\left[-\frac{7\pi }{4},\frac{\pi }{4}\right]$ and $\left[\frac{\pi }{4},\frac{9\pi }{4}\right]$ .