The domain is indeed all real numbers, since you can put any real number in for x and get an f(x). (That is, you can take the sine and cosine of any real number, add those two numbers together, and get a real number.)
range of sinx= -1 to 1.. so |sinx|=0 to 1
similarly for cosx= -1 to 1..so |cosx|= 0 to 1
therefore range of |sinx|+|cosx|= will never be 0 to 2 coz This is because sin(x) and cos(x) never both equal 1 or -1 at the same time,
f(x)=f(0)= sin 0 +cos 0=1
f(30)= 1/2+root(3)/2=1.366
f(45)= 1/root(2)+1/root(2)=1.414
f(60)= 1.366
f(90)= sin90+cos 90=1
and the other domains will also map to this values only
so range ={1,1.366,1.414}