Cobweb plot for a function off of StackExchange

Can anyone show me what a cobweb plot for
$$f(x) = \sin(x) + x$$
looks like??

This came from this very fun StackExchange question.

This shows an attractive iteration at 5.5 going to Pi. If I went past 6.3, It would then be attracted towards 3Pi.

Here's the cobweb plot if we use $x_0=1$:

Notice how it converges to the fixed point $π$

As long as we continue to use a positive point $x_0<2π$, the plot will continue to converge to the fixed point $π$, such as here where I used a point $x_0=6$:

As Keith mentioned, using a point $x_0>2π$ (about $6.283$) in this case will result in the plot converging towards a different fixed point $3π$, as I've demonstrated here:

For this cobweb plot I used the point $x_0=7$, and we can see how the plot is now converging towards the fixed point $3π$ (about $9.425$)