Chapter 4, Problem 23RE

To determine

The features of the family members of the curve.

Explanation of Solution

Given:

The function is

  $\mathrm{f}\left(\mathrm{x}\right)=\ln \left(\sin \mathrm{x}+\mathrm{C}\right)$

Concept used:

The function is periodic with period of $2\mathrm{\pi }$ with maximum at $2\mathrm{n}\mathrm{\pi }+\frac{\mathrm{\pi }}{2}$ .

Calculation:

The function is

  $\mathrm{f}\left(\mathrm{x}\right)=\ln \left(\sin \mathrm{x}+\mathrm{C}\right)\mathrm{................................}\left(1\right)$

The graph function will undefined

If $\mathrm{C}\le -1$ because $\left(\sin \mathrm{x}+\mathrm{C}\right)$ will become zero or negative and the natural logarithmic of zero or a negative number is undefined.

Using the value $\mathrm{C}=0,1,2\mathrm{a}\mathrm{n}\mathrm{d}3$

  $\begin{array}{l}\mathrm{y}=\ln \left(\sin \mathrm{x}+0\right)\\ \mathrm{y}=\ln \left(\sin \mathrm{x}+1\right)\\ \mathrm{y}=\ln \left(\sin \mathrm{x}+2\right)\\ \mathrm{y}=\ln \left(\sin \mathrm{x}+3\right)\end{array}$

The graph that the function is periodic with period of $2\mathrm{\pi }$ with maximum at $2\mathrm{n}\mathrm{\pi }+\frac{\mathrm{\pi }}{2}$

For the function to be continuous on $\left(-\mathrm{\infty },\mathrm{\infty }\right)$ the function has to be defined for any value of $\mathrm{x}$

The function will have no graph when $\mathrm{C}\le -1$

When $\mathrm{C}\to \mathrm{\infty }$ the function becomes a straight line parallel to $\mathrm{x}-\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}$

Draw the graph

  $\mathrm{f}\left(\mathrm{x}\right)=\ln \left(\sin \mathrm{x}+\mathrm{C}\right)$