Figure.(1) shows the plot of the function f(x) = x + sin(x) along with the plot of function f(x) = x .
Obviously the function f(x) = x+sin(x) is overlap of sin(x) on the line of f(x) =x .
The inverse function of x+sin(x) is defined as f(x) = x-sin(x). It is just definition. It is not calculated using any guidelines.
Inverse function x-sin(x) is plotted in figure. It can be seen from figure.(2) and figure. (3) the function x+sin(x) and
its inverse are mutual reflection of other with respect to the line f(x) = x. It appears mirror is placed along the line f(x) = x
and both the fuctions are getting reflected. That is why the function ' x-sin(x) ' is called the inverse of ' x+sin(x) ' and vice versa.
Area enclosed between the two functions (Shaded area in figure 3) =