Cosine Function Domain Restriction Options?

Topic:
Cosine
Recall that in order for a relation to be a function, each & every input can have one and only one output. The cosine function, part of whose graph is displayed below, is a function because each input (angle) can only have one cosine ratio to which it maps (see this applet by Anthony C.M. Or). Interact with the applet below for a minute, then answer the questions that follow.
Directions: Click the refresh (recycle) icon at the top of the applet. Then select the Show Inverse Relation checkbox. Drag the x-final slider slowly to the right and note the inverse relation being graphed simultaneously as the cosine function is being graphed.

1.

Why is the relation considered to be a function?

2.

Select the Default to Natural Domain of f checkbox. Then select Show Inverse Relation. Is this inverse relation (whose equation is ) also a function? Explain why or why not.

3.

How could we restrict the domain of the original function in order for its inverse relation to be a function as well? There are lots of possibilities. Can you find one? Feel free to experiment by inputting values into the Xmin and Xmax input boxes or by using the sliders.)