Question

Can these functions be defined for all real numbers?

y= e^cos(x)
y= ln(sin(x))
y= sqr(cos(x)+2)

I'm trying to find some systematic way to test these, but I'm not sure how I can proe it for any possible input.

Answer 1

cos is defined for all reals, so e^cos is too

ln is defined for x>0, so we need sin > 0; x must be in QI or QII

sqr is defined for x>=0, so we need cos(x)+2 >= 0. since cosx >= -1, cos+2 >= 1 for all x, so this one is also defined for all reals.