Express r=2cos (theta) + 2sin (theta) as a cartesian function.

The function is in polar coordinates.

When this is the case, to pass to rectagular (cartesian) coordinates you use:

x = r cos (theta)

y = r sin (theta)

Then,

x = [2cos (theta) + 2sin (theta) ]cos (theta) =

= 2 [cos (theta) ]^2 + 2sin (theta) cos (theta) = 2 [cos (theta) ]^2 + sin (2theta)

y = [2cos (theta) + 2sin (theta) ]sin (theta) =

= 2 cos (theta) sin (theta) + 2[sin (theta) ]^2 = sin (2theta) + 2[sin (theta) ]^2