Final answer:
The cell cortex is responsible for allowing cells to change shape and move. It also supports and strengthens the cell membrane, but does not influence the passage of molecules or restrict protein movement.
Step-by-step explanation:
The cell cortex is a specialized layer of cytoplasmic material that lies just beneath the cell membrane and helps maintain the shape and physical characteristics of the cell. In particular, the cell cortex is composed of an actin filament network which is important for maintaining the cell's shape, enabling changes in cell shape, and facilitating cellular movement. The correct function of the cell cortex as outlined in the options provided is that it allows cells to change shape and to move (b).
The cell cortex supports and strengthens the membrane (e), influencing the mechanical properties of the cell but it does not influence the passage of small molecules into and out of the cell (a), as that function primarily belongs to the plasma membrane itself. The cell cortex does not lubricate the cell (c) or restrict the movement of certain proteins in the lipid bilayer (d), since these are not functions commonly attributed to the cortex layer.