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 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}

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2 votes

Domain = R  
Range = 0<=|sin x|<=1, 
               0<=|cos x|<=1. 
Graph of f(x) will have maximum values at pi/4, 3pi/4.... with the maximum value of 1.414 and minimum value of F(x) is 0, at 0, pi/2.... 
Thus range lies between [1,1.414].

here is the graph of f(x)

It can be also be obtain by differentiating f(x) w.r.t. x , then equating it to 0 and finding the value of x and get the graphs max and min value, which is nothing but range of the graph.

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